PUBLICATIONS 

OF  THE  POLLAK  FOUNDATION  FOR 
ECONOMIC  RESEARCH 


NUMBER  ONE 
THE  MAKING  OF  INDEX  NUMBERS 


THE  MAKING 
OF  INDEX  NUMBERS 

A  Study  of  Their 
Varieties,  Tests,  and  Reliability 

BY 
IRVING  FISHER 

PROFESSOR  OF  POLITICAL  ECONOMY,  TALE  UNIVERSITY 


BOSTON  AND  NEW  YORK 

HOUGHTON  MIFFLIN  COMPANY 

fctje  ftifaersibe  $)tess  Camfrifcge 
1922 


COPYRIGHT,   IQ22,  BY  THE  POLLAK  FOUNDATION  FOR  ECONOMIC  RESEARCH 
ALL  RIGHTS  RESERVED 


tEfjt  »toet*toe 

CAMBRIDGE  •  MASSACHUSETTS 
f  RINTED  IN  THE  U.S.A. 


TO 

F.  Y._EDGEWORTH 

AND 

CORREA  MOYLAN  WALSH 

PIONEERS  IN  THE 
EXPLORATION  OF  INDEX  NUMBERS 


49?U.'J3 


PREFATORY  NOTE 

ALL  sciences  are  characterized  by  a  close  approach  to  exact 
measurement.  How  many  of  them  could  have  made  much 
progress  without  units  of  measurement,  generally  under- 
stood and  accepted,  it  is  difficult  to  imagine.  In  order  to 
determine  the  pressure  of  steam,  we  do  not  take  a  popular 
vote :  we  consult  a  gauge.  Concerning  a  patient's  tempera- 
ture, we  do  not  ask  for  anybody's  opinion:  we  read  a  ther- 
mometer. In  economics,  however,  as  hi  education,  though 
the  need  for  quantitative  measurement  is  as  great  as  hi 
physics  or  hi  medicine,  we  have  been  guided  hi  the  past 
largely  by  opinions  and  guesses.  In  the  future,  we  must 
substitute  measurement  for  guesswork.  Toward  this  end, 
we  must  first  agree  upon  uistruments  of  measurement.  To 
the  Pollak  Foundation  for  Economic  Research  it  seems 
fitting,  therefore,  that  its  first  publication  should  be  The 
Making  of  Index  Numbers. 

In  this  book,  the  author  tests  by  every  useful  method, 
not  only  all  the  formulae  for  index  numbers  that  have  been 
used,  but  as  well  all  that  reasonably  could  be  used;  and  he 
tests  them  by  means  of  actual  calculations,  extensive  and 
painstaking,  based  on  actual  statistical  records.  He  proves 
that  several  of  the  methods  of  constructing  index  numbers 
now  hi  common  use  are  grossly  inaccurate;  he  makes  clear 
why  some  formulae  are  precise  and  others  far  from  it;  he 
points  out  how  to  save  tune  in  the  work  of  calculation;  and 
he  shows  how  to  test  the  results.  Thus  he  provides  us  with 
methods  of  measuring  such  illusive  things  as  fluctuations 
in  real  wages,  in  exchange  rates,  in  volume  of  trade,  in  the 
cost  of  living,  and  in  the  purchasing  power  of  the  dollar. 
Finally,  he  points  out  that,  once  a  good  method  of  con- 


viii  PREFATORY  NOTE 

structing  index  numbers  has  been  generally  accepted,  the 
usefulness  of  the  instrument  will  be  vastly  increased,  and 
will  then  be  extended  to  many  other  fields  where  precise 
measurement  is  greatly  needed. 

But,  after  all,  is  it  possible  to  devise  a  means  of  measure- 
ment that  is  sufficiently  precise  to  be  used  as  a  basis  for 
determining  matters  of  such  concern  to  all  human  beings 
as  contracts,  currency  measures,  price  adjustments,  and 
wage  schedules?  The  doubts  on  this  question  that  have 
hitherto  stood  in  the  way  of  the  universal  use  of  index 
numbers  must  vanish  before  Professor  Fisher's  demonstra- 
tions. He  shows  that  an  index  number  may  be  so  precise 
an  instrument  that  the  error  "  probably  seldom  reaches  one 
part  in  800,  or  a  hand's  breadth  on  the  top  of  Washington 
Monument,  or  less  than  three  ounces  on  a  man's  weight,  or 
a  cent  added  to  an  $8  expense."  He  shows,  further,  that  all 
the  forms  of  index  numbers  that  satisfy  his  few,  simple 
tests  give  results  so  nearly  alike  that  it  matters  little  or 
nothing,  for  most  practical  purposes,  which  form  is  em- 
ployed. Any  one  of  these  forms  is  comparable,  in  point  of 
accuracy,  with  many  of  the  instruments  that  are  univer- 
sally and  unquestioningly  employed  in  other  scientific 
fields. 

The  use  of  yardsticks  of  forty  different  lengths  would  be 
a  source  of  endless  confusion:  the  use  of  forty  different 
kinds  of  index  numbers  is  no  less  confusing.  If  experts  fail 
to  clear  up  this  confusion  because  they  disagree  on  non- 
essentials,  it  will  seem  to  the  many  thousands  of  people, 
to  whom  the  mathematics  of  the  subject  is  a  mystery,  as 
though  the  experts  were  widely  at  variance  on  fundamen- 
tals. And  so,  without  due  cause,  index  numbers  in  general 
will  be  discredited  and  the  scientific  study  of  economics 
impeded.  For  this  reason,  it  is  to  be  hoped  that  all  those 
who  are  capable  of  understanding  the  subject  will  see  their 


PREFATORY  NOTE  ix 

way  clear  to  agreeing  upon  the  Ideal  Formula  as  the  best 
in  point  of  accuracy.  It  is  to  be  hoped,  furthermore,  that 
they  will  agree  in  adopting  and  advocating  for  general  use 
the  closely  similar  Formula  No.  2153 


vS(<?c 

since  it  is  the  one  which  best  combines  speed  of  calculation 
with  as  high  a  degree  of  accuracy  as  is  ever  needed  for  prac- 
tical purposes.  In  any  event,  the  Pollak  Foundation  will 
have  achieved  its  purpose  in  publishing  this  volume,  if  it 
has  a  part  in  bringing  about  the  abandonment  of  faulty 
methods  of  constructing  index  numbers,  the  general  adop- 
tion of  any  dependable  method,  and  the  consequent  pro- 
gressive substitution,  wherever  precise  measurement  is 
possible,  of  scientific  method  for  personal  opinion. 

WILLIAM  TRUFANT  FOSTER 

Editor  of  the  Pollak  Publications 
NEWTON,  MASSACHUSETTS 
December  1,  1922 


PREFACE 

THIS  book  amplifies  a  paper  read  in  December,  1920,  at 
the  Atlantic  City  meeting  of  the  American  Statistical 
Association.  An  abstract  of  that  paper  was  printed  in  1921 
in  the  March  number  of  the  Association's  Quarterly  Pub- 
lication. The  same  paper,  somewhat  elaborated,  was  also 
read  before  the  American  Academy  of  Arts  and  Sciences  at 
Boston,  in  April,  1921. 

One  of  the  main  conclusions  of  these  papers  was  accepted 
at  once,  namely,  that  the  formula  here  called  the  " ideal" 
is  the  best  form  of  index  number  for  general  purposes. 
The  further  contention  that  this  formula  is  the  best  for  all 
purposes  was  stoutly  denied  by  most  critics,  with  the  not- 
able exception  of  Mr.  C.  M.  Walsh,  who  had  reached  the 
same  conclusion  independently  and  from  a  different  start- 
ing point. 

Out  of  this  partial  disagreement,  a  number  of  writings 
on  index  numbers  have  appeared,  such  as  Professor  War- 
ren M.  Persons'  article  in  the  Review  of  Economic  Statis- 
tics for  May,  1921,  on  "  Fisher's  Formula  for  Index  Num- 
bers." Professor  Allyn  A.  Young  in  his  article  "The  Meas- 
urement of  Changes  of  the  General  Price  Level,"  in  the 
Quarterly  Journal  of  Economics  for  August,  1921,  reaches 
the  same  formula  as  "the  best  single  index  number  of  the 
general  level  of  prices,"  although  he  apparently  reserves 
judgment  as  to  its  limitations.  Professor  Wesley  C.  Mit- 
chell, in  the  revision  of  his  monograph  on  "Index  Num- 
bers of  Wholesale  Prices  in  the  United  States  and  Foreign 
Countries  "  (published  as  Bulletin  No.  284  of  the  United 
States  Bureau  of  Labor  Statistics,  October,  1921),  takes 
a  somewhat  similar  position. 


xii      •  PREFACE 

In  order  to  help  resolve  the  questions  remaining  at  issue, 
a  large  number  of  calculations  have  been  made  for  this 
book  in  addition  to  the  large  number  which  had  already 
been  made.  Any  one  who  has  not  himself  attempted  a 
like  task  can  scarcely  realize  the  amount  of  time,  labor, 
and  expense  involved.  Some  of  the  work  must  have  been 
abandoned  had  not  the  Pollak  Foundation  for  Economic 
Research  come  to  the  rescue. 

The  result  has  been  a  much  more  complete  survey  of 
possible  formulae  than  any  hitherto  attempted.  Although, 
in  a  subject  like  this,  absolute  completeness  is  out  of  the 
question,  since  the  number  of  possible  formulas  is  infinite, 
nevertheless,  the  whole  field  has  been  so  mapped  out  as  to 
leave  no  large  gaps.  The  aim  has  been  to  settle  decisively, 
if  possible,  the  questions  of  how  widely  the  various  results 
reached  by  different  possible  methods  diverge  from  each 
other,  and  why.  Each  of  more  than  a  hundred  formulae 
has  been  examined  and  calculated  in  four  series.  Each  of 
these  series  has  its  role  to  play  in  this  study,  even  formulae 
which  are  found,  in  the  end,  to  have  no  practical  use. 

This  book  is,  therefore,  primarily  an  inductive  rather 
than  a  deductive  study.  In  this  respect  it  differs  from 
the  Appendix  to  Chapter  X  of  the  Purchasing  Power  of 
Money,  in  which  I  sought  deductively  to  compare  the 
merits  of  44  different  formulae.  The  present  book  had  its 
origin  in  the  desire  to  put  these  deductive  conclusions  to 
an  inductive  test  by  means  of  calculations  from  actual  his- 
torical data.  But  before  I  had  gone  far  in  such  testing  of  my 
original  conclusions,  I  found,  to  my  great  surprise,  that  the 
results  of  actual  calculation  constantly  suggested  further 
deduction  until,  in  the  end,  I  had  completely  revised  both 
my  conclusions  and  my  theoretical  foundations.  Not  that 
I  needed  to  discard  as  untrue  many  of  the  conclusions 
reached  in  the  Purchasing  Power  of  Money,  for  the  only 


PREFACE  xiii 

definite  error  which  I  have  found  among  my  former  con- 
clusions has  to  do  with  the  so-called  " circular  test"  which 
I  originally,  with  other  writers,  accepted  as  sound,  but 
which,  in  this  book,  I  reject  as  theoretically  unsound.  But 
some  of  the  other  tests  given  in  the  Purchasing  Power  of 
Money,  while  perfectly  legitimate,  are  of  little  value  as 
quantitative  criteria  for  a  good  index  number.  The  most 
fundamentally  important  test  among  those  treated  in  the 
earlier  study  is  the  "time  reversal"  test.  This  and  a  new 
test,  the  "factor  reversal"  test,  are  here  constituted  the 
two  legs  on  which  index  numbers  can  be  made  to  walk. 

In  the  algebraic  analysis,  relegated  almost  wholly  to 
the  Appendices,  I  have  refrained,  as  far  as  possible,  from 
developing  the  many  possible  mathematical  transforma- 
tions and  discussions  of  index  numbers,  in  the  belief  that, 
fascinating  as  these  are,  the  mathematics  of  index  numbers, 
except  as  they  serve  practical  ends,  would  not  interest 
many  readers.  For  the  same  reason  such  mathematical 
analysis  as  is  included  has,  in  some  cases,  been  greatly 
condensed,  the  results  alone  being  given.  Without  such 
condensation  of  unimportant  details  a  hundred  or  more 
additional  pages  in  the  Appendices  would  have  been 
necessary. 

One  incidental  result  of  this  study  is  to  show  that  many 
precise  and  interesting  relations  or  laws  exist  connecting 
the  various  magnitudes  studied  —  index  numbers,  dis- 
persions, bias,  correlation  coefficients,  etc.  Thus  this  field 
of  study,  almost  alone  in  the  domain  of  the  social  sciences, 
may  truly  be  called  an  exact  science  —  if  it  be  permissible 
to  designate  as  a  science  the  theoretical  foundations  of  a 
useful  art. 

The  subject  has  seemed  elusive  because  it  is  partly  em- 
pirical and  partly  rational,  and  these  two  aspects  of  it  have 
not  been  coordinated.  But,  although  the  present  volume 


xiv  PREFACE 

is  a  combination  of  theoretical  and  practical  discussion,  the 
theoretical  is  entirely  in  the  interest  of  the  practical.  Most 
writers  on  index  numbers  have  been  either  exclusively  the- 
oretical or  exclusively  practical,  and  each  of  these  two 
classes  of  writers  has  been  very  little  acquainted  with  the 
other.  By  bringing  these  two  worlds  into  closer  contact 
I  hope  that,  in  some  measure,  I  may  have  helped  forward, 
both  the  science  and  the  art  of  index  numbers.  The  im- 
portance of  this  new  art  in  our  economic  life  is  already 
great  and  is  rapidly  increasing. 

While  the  book  includes,  I  hope,  all  the  chief  results  of 
former  studies  in  index  numbers,  its  main  purpose  is  not 
so  much  to  summarize  previous  work  as  to  add  to  our 
knowledge  of  index  numbers  and,  as  a  consequence,  to  set 
up  demonstrable  standards  of  the  accuracy  of  index  num- 
bers and  their  suitability  under  various  circumstances. 
Many  of  the  results  reached  turned  out  to  be  quite  different 
from  what  I  had  been  led  by  the  previous  studies  of  others 
as  well  as  of  myself  to  expect. 

I  am  greatly  indebted  to  the  many  persons  who  have 
helped  me  in  the  preparation  of  the  book  —  especially  to 
Mr.  R.  H.  Coats,  Dominion  Statistician  of  Canada;  Dr. 
Royal  Meeker,  Chief  of  the  Scientific  Division  of  the  Inter- 
national Labour  Office;  Professor  Warren  M.  Persons,  of 
the  Harvard  Committee  on  Economic  Research;  Pro- 
fessor Allyn  A.  Young,  of  Harvard  University;  Professor 
Wesley  C.  Mitchell,  of  Columbia  University;  Director 
William  T.  Foster  and  Professor  Hudson  Hastings,  of 
the  Pollak  Foundation  for  Economic  Research;  Professor 
Frederick  R.  Macaulay,  of  the  National  Bureau  of  Eco- 
nomic Research;  Mr.  Correa  Moylan  Walsh;  Mr.  H.  B. 
Meek,  instructor  at  Yale  in  mathematics;  Mr.  V.  I.  Caprin, 
Mr.  L.  B.  Haddad  and  Mr.  M.  H.  Wilson,  Yale  students; 
Miss  Else  H.  Dietel,  my  research  secretary,  and  my  brother, 


PREFACE  xv 

Mr.  Herbert  W.  Fisher.  Professor  Hastings,  Mr.  Meek, 
Mr  Walsh,  Miss  Dietel,  and  my  brother  have  read  the 
entire  manuscript.  Their  very  valuable  criticisms  and  sug- 
gestions have  been  both  detailed  and  general. 

IRVING  FISHER 

YALE  UNIVERSITY 
October,  1922 


Addendum  to  pp.  240-242. 

As  this  book  goes  to  press,  Professor  Allyn  A.  Young,  of 
Harvard  University,  writes,  calling  attention  to  the  fact 
that  what  I  call  the  "  ideal  formula "  is  mentioned  by  Ar- 
thur L.  Bowley  in  Palgrave's  Dictionary  of  Political  Econ- 
omy, vol.  in,  p.  641.  This  reference,  in  1899,  antedates  any 
of  those  mentioned  on  p.  241  in  this  book. 

Referring  to  measuring  the  "aisance  relative,"  or  the 
relative  well-being  of  labor,  Bowley  says:  "The  best 
method  theoretically  for  measuring  'aisance  relative'  ap- 
pears to  be  as  follows:  calculate  the  quantity  by  method 
(ii)  twice,  taking  first  a  budget  typical  of  the  earlier,  then 
of  the  later  year,  valuing  them  at  the  prices  of  both  years 
and  obtaining  two  ratios.  The  average  (possibly  the  geo- 
metric rather  than  the  arithmetic)  of  these  ratios  measures 
the  relative  'aisance/"  He  then  gives,  among  others,  the 
"ideal  formula." 


SUGGESTIONS  TO  READERS 

IN  GENERAL 

AFTER  finishing  one  of  his  long  and  much  worked-over 
novels,  Robert  Louis  Stevenson  expressed  a  fear  that  no- 
body would  read  it.  In  these  days  of  many  books  and 
little  time  for  reading  them,  when  even  a  novel  must  be 
short  to  be  much  read,  a  book  on  Index  Numbers  can 
hardly  expect  to  become  a  "best  seller."  This  book  has 
grown  to  three  times  the  length  originally  contemplated, 
and  the  very  effort  to  make  it  readable  for  the  general 
reader  has  increased  its  length. 

It  aims  to  meet  the  needs  of  several  quite  different 
classes  of  readers:  the  specialist  hi  index  numbers  who  is 
mathematical;  the  specialist  who  is  not  mathematical;  the 
university  student  who  would  become  a  specialist;  the 
student  who  wants  merely  to  understand  the  fundamental 
principles  of  the  subject;  the  practical  user  of  index  num- 
bers; and,  finally,  the  general  reader  —  he  who  merely 
wants  to  know  something  about  index  numbers.  The  book 
is  also  intended  to  serve  as  a  reference  book  to  be  consulted 
by  economists,  and  statisticians,  including  business  statis- 
ticians. It  is  also  designed  to  serve  as  a  textbook  for 
classes  in  statistics.  While  it  aims  primarily  to  add  to  our 
knowledge  of  index  numbers  and  to  set  new  standards  for 
judging  them,  most  of  the  time  and  effort  expended  upon 
the  writing  have  been  directed  toward  the  reader  who  has 
had  no  previous  acquaintance  with  the  subject.  In  other 
words,  the  book  tries  to  popularize  the  exposition  even  of 
the  somewhat  intricate  parts  which  are  placed  in  the  appen- 
dix. The  body  of  the  text,  especially  if  the  fine  print  be 
omitted,  ought  to  be  intelligible  to  any  intelligent  reader. 


xviii  SUGGESTIONS  TO  READERS 

Every  important  point  has  been  illustrated  graphically. 
There  are  123  charts.  I  believe  that  the  chief  reason  why, 
hitherto,  the  making  of  index  numbers  has  been  a  mys- 
tery to  most  people  is  the  absence  of  just  such  graphic 
charts. 

IN  PARTICULAR 

1.  Only  the  specialist  in  the  field  of  Index  Numbers  is 
expected  to  read  every  word.    Appendix  1,  consisting  of 
notes  on  the  text,  is  best  read  in  conjunction  with  the  pas- 
sages to  which  the  notes  relate. 

2.  The  non-mathematical  reader  will  doubtless  omit  the 
mathematical  parts  of  the  Appendix.    He  need  not  omit 
the  few  simple  algebraic  expressions  in  the  text,  nor  all  of 
the  appendix  material. 

3.  The  non-specialist  may  well  omit  all  the  appendix 
material,   although  the  non-mathematical  parts  of  the 
Appendix  are  almost  as  easy  to  read  as  the  text,  having 
been  placed  in  the  Appendix  merely  because  concerned 
with  details  or  side  issues. 

4.  The  general  reader  who  would  still  further  shorten 
the  time  of  reading  may  omit  the  fine  print,  reducing  the 
printed  pages  to  be  read  to  216  exclusive  of  the  71  pages 
of  diagrams  and  the  33  pages  of  tables.  He  may  also  omit 
the  system  of  paragraphs  occurring  in  almost  every  section 
after  the  italicized  words,  "numerically"  and  "algebrai- 
cally," totalling  20  pages,  unless,  after  reading  a  paragraph 
beginning  with  the  word  "graphically,"  he  feels  the  need 
of  supplementing  such  paragraphs  by  the  numerical  or  al- 
gebraic expressions.    These  three  methods  of  exposition, 
numerical,  graphic,  and  algebraic,  run  parallel  through- 
out the  book.  This  reduces  the  number  of  pages  to  196. 

5.  The  mere  skimmer  will  find  the  main  conclusions  hi 
the  last  chapter,  XVII. 


SUGGESTIONS  TO  READERS  xix 

6.  The  use  of  the  book  as  one  of  reference  will  be  facili- 
tated by  the  list  of  tables  and  charts,  the  "keys"  re- 
ferred to  below,  the  section  headings,  the  italicized  words 
"numerically"  "graphically"  "algebraically"  and  the  ar- 
rangement of  the  charts  which,  when  they  occur  in  pairs, 
places  the  price  chart  always  on  the  left  page  and  the 
quantity  chart  on  the  right. 

7.  Appendix  V  may  be  referred  to  with  profit  whenevet 
the  reader  finds  mention  of  a  formula  merely  by  its  iden- 
tification number.    Attention  is  especially  called  to  §  1 
and  §  2  of  Appendix  V,  the  "Key  to  the  Principal  Alge- 
braic Notations/'  and  the  "Key  to  Numbering  of  For- 
mulae." A  few  minutes'  inspection  of  this  easy  mnemonic 
system  of  numbering  will  enable  the  reader  to  recognize 
at  sight  "Formula  1,"  "Formula  21,"  "Formula  53," 
"Formula  353,"   or  any  other  number,  and  mentally 
locate  it  in  the  system. 

8.  The  reader  seeking  directions  for  computing  index 
numbers  by  the  nine  most  practical  formulae  will  find 
them  in  Appendix  VI,  §  2. 


TABLE  OF  CONTENTS 

PAGE 

INTRODUCTION ,1 

Six  TYPES  OF  INDEX  NUMBERS  COMPARED     .      .      .      .11 

FOUR  METHODS  OF  WEIGHTING 43  -^ 

Two  GREAT  REVERSAL  TESTS 62 

'  -.        I      -— 

V.  ERRATIC,  BIASED,  AND  FREAKISH  INDEX  NUMBERS     .  .    83 

VI.  THE  Two  REVERSAL  TESTS  AS  FINDERS  OF  FORMULAE   .  .118 

VII.  RECTIFYING  FORMULA  BY  "CROSSING"  THEM       .      .  .136 

VIII.  RECTIFYING  FORMULAE  BY  CROSSING  THEIR  WEIGHTS  .  .  184 

IX.  THE  ENLARGED  SERIES  OF  FORMULA 197 

1.  WHAT  SIMPLE  INDEX  NUMBER  is  BEST?    .      .      .      .  .  206 
XI.  WHAT  is  THE  BEST  INDEX  NUMBER?         .... 

XII.  COMPARING  ALL  THE  INDEX  NUMBERS  WITH  THE  "  IDEAL/" 

(FORMULA  353) 243 

XIII.  THE  SO-CALLED  CIRCULAR  TEST \  270 

"J.V.  BLENDING  THE  APPARENTLY  INCONSISTENT  RESULTS  .      .  297 

XV.  SPEED  OF  CALCULATION 321 

XVI.  OTHER  PRACTICAL  CONSIDERATIONS 330 

XVII.  SUMMARY  AND  OUTLOOK 350 

APPENDIX      I.  NOTES  TO  TEXT       . 371 

APPENDIX     II.  THE  INFLUENCE  OF  WEIGHTING       ....  439 

APPENDIX   III.  AN  INDEX  NUMBER  AN  AVERAGE  OF  RATIOS 

RATHER  THAN  A  RATIO  OF  AVERAGES    .  .451 


XX11 


TABLE  OF  CONTENTS 


PAGE 

APPENDIX  IV.  LANDMARKS  IN  THE  HISTORY  OF  INDEX  NUMBEBS  458 
APPENDIX  V.  LIST  OP  FORMULA  FOR  INDEX  NUMBERS  .  .  461 
APPENDIX  VI.  NUMERICAL  DATA  AND  EXAMPLES  .  .  .  489 

APPENDIX  VII.  INDEX  NUMBERS  BY  134  FORMULA  FOR  PRICES 
BY  THE  FIXED  BASE  SYSTEM  AND  (IN  NOTE- 
WORTHY CASES)  THE  CHAIN  SYSTEM  .  .  .  498 

APPENDIX  VIII.  SELECTED  BIBLIOGRAPHY 519 

INDEX  .  521 


LIST  OF  TABLES 
CHAPTER  II 

TABLE  PAGE 

1.  The  Simple  Arithmetic  Index  Number  for  1914  as  Calculated 
from  the  36  Prices  for  1913  and  1914       ......     16 

2.  Simple  Arithmetic  Index  Number  (Formula  1)  for  Prices       .    23 

CHAPTER  III 

3.  Values  (in  Millions  of  Dollars)  of  Certain  Commodities    .      .  47 

4.  Hybrid  Values  (in  Millions  of  Dollars)  of  Certain  Commodi- 
ties ...............  55 

5.  Identification  Numbers  of  Primary  Formulae        .      ...  61 

CHAPTER  IV 

6.  Value  Ratios  for  36  Commodities,  1913-1918     ....    74 

CHAPTER  V 

7.  Joint  Errors  of  the  Forward  and  Backward  Applications  of 
Each  Formula  (that  is,  under  Test  1)  in  Per  Cents    ...     84 

8.  Joint  Errors  of  the  Price  and  Quantity  Applications  of  Each 
Formula  (that  is,  under  Test  2)  in  Per  Cents       ....    85 

9.  The  Four  Systems  of  Weighting  the  Price  Relatives  for  1917, 


97 


10.  Dispersion  of  36  Price  Relatives,  (1)  before  the  World  War, 
and,  (2)  during  it     .........      .      .  109 

11.  Dispersion  of  36  and  of  12  Quantity  Relatives      .      .      .      .110 

CHAPTER  VII 

12.  Identification  Numbers  of  Formulae  of  Main  Series    .      .      .  182 

CHAPTER  VIII 

13.  Derivation  of  Cross  Formulas  and  Cross  Weight  Formulae      .   186 

14.  Index  Numbers  by  Cross  Weight  Formulae  (1123,  1133,  1143, 


xxiv  LIST  OF  TABLES 

TABLE  PAGE 

1153)    compared   with   Index   Numbers    by   Corresponding 
Cross  Formulae  (123,  133,  143,  153) 189 

15.  Index  Numbers  by  Cross  Weight  Formulae  (1003,  1004,  1013, 
1014)  compared  with  Corresponding  Cross  Formulas  .      .      .  190 

16.  Doubly  Rectified  Formulae  derived  from  Primary  Weighted 
Formulae 194 

CHAPTER  IX 

17.  Enlarged  Arithmetic-Harmonic  Group     .      .      .      .      .      .  199 

18.  Enlarged  Geometric  Group 200 

19.  Enlarged  Median  Group 200 

20.  Enlarged  Mode  Group 201 

21.  Enlarged  Aggregative  Group 201 

22.  The  Five-Tined  Fork 204 

CHAPTER  XI 

23.  Excess  or  Deficiency  of  Simple  Mode  of  Price  Relatives  (in  Per 
Cents  of  Simple  Geometric) 216 

24.  Excess  or  Deficiency  of  Simple  Median  of  Price  Relatives  (in 
Per  Cents  of  Simple  Geometric) 217 

25.  Two  Best  Index  Numbers 224 

26.  Selected  Index  Numbers 226 

27.  Probable  Errors  of  an  Index  Number  of  Prices  or  Quantities 
worked  out  by  anyone  of  the  13  Formulae  considered  as  Equally 
Good  Independent  Observations 227 

CHAPTER  XII 

28.  Index  Numbers  by  134  Formulae  arranged  in  the  Order  of  Re- 
moteness from  the  Ideal  (353)  (as  shown  by  the  Fixed  Base 
Figures  for  the  Price  Indexes) 244 

29.  Averages  of  Each  of  the  Various  Classes  of  Index  Numbers    .  257 

30.  Accuracy  of  Simple  Geometric  and  Simple  Median,  judged  by 
the  Standard  of  Formula  353  for  36  Commodities,  Prices    .      .  261 

31.  Accuracy  of  Simple  Geometric  and  Simple  Median,  judged  by 
the  Standard  of  Formula  353  for  36  Commodities,  Quantities  .  261 

32.  Accuracy  of  Simple  Geometric  and  Simple  Median,  judged  by 
the  Standard  of  Formula  53  for  1437  Commodities  .  262 


LIST  OF  TABLES  xxv 

CHAPTER  XIII 

TABLE  PAGE 

33.  The  "Circular  Gap,"  or  Deviation  from  fulfilling  the  So-called 
"Circular  Test"  of  Various  Formulae 278 

34.  The  "Circular  Gap,"  or  Deviation  from  fulfilling  the  So-called 
"Circular  Test,"  of  Formula  353  (in  all  possible  3-Around 
Comparisons  of  Price  Indexes) 280 

35  The  "Circular  Gap,"  or  Deviation  from  fulfilling  the  So-called 
"Circular  Test,"  of  Formula  353  (in  all  possible  4- Around 
Comparisons  of  Price  Indexes) 281 

36.  The  "Circular  Gap,"  or  Deviation  from  fulfilling  the  So-called 
"Circular  Test,"  of  Formula  353  (in  all  possible  5- Around 
Comparisons  of  Price  Indexes) 282 

37.  The  "Circular  Gap,"  or  Deviation  from  fulfilling  the  So-called 
"Circular  Test,"  of  Formula  353  (in  all  possible  6- Around 
Comparisons  of  Price  Indexes) 283 

38.  "Circular  Gaps  "for  Formula  353 284 

39.  List  of  Formulae  in  (Inverse)  Order  of  Conformity  to  So-called 
Circular  Test ; 289 

CHAPTER  XIV 

40.  Formula  353  on  Bases  1913, 1914,  1915, 1916,  1917, 1918;  also 
Formula  7053,  the  Average  of  the  Six  Preceding,  and  7053  re- 
duced to  make  the  1913  Figure  100 301 

41.  Four  Single  Series  of  Six  Index  Numbers  as  Makeshifts  for  the 
Complete  Set  of  Table  40,  Prices 307 

42.  Four  Single  Series  of  Six  Index  Numbers  as  Makeshifts  for  the 
Complete  Set  of  Table  40,  Quantities 307 

43.  The  Influence  of  Broadening  the  Base  in  reducing  Bias    .      .315 

CHAPTER  XV 

44.  Rank  in  Speed  of  Computation  of  Formulas 322 

CHAPTER  XVI    . 

45.  Deviations  from  200  Commodities  Index 337 

46.  Comparison  of  the  Aggregate  Value  of  the  100,  50,  25,  12,  6, 
and  3  Commodities  with  the  Aggregate  Value  of  the  200  Com- 
modities        339 

47.  (Inverse)  Order  of  Rank  of  Formulas       .  .  348 


xxvi  LIST  OF  TABLES 

APPENDIX  I  (NOTE  TO  CHAPTER  V,  §  11) 

TABLE  PAGE 

48.  For  Finding  the  Bias  corresponding  to  any  given  Dispersion  390 

49.  Standard  Deviations  (for  Prices) 391 

50.  Special  Dispersion  Index  compared  with  Standard  Deviation 
(logarithmically  calculated)  for  the  36  Commodities  (Simple) .  392 

51.  Special  Dispersion  Index  compared  with  Standard  Devia- 
tion  (logarithmically   calculated)  for  the  36  Commodities 
(Weighted) 393 

APPENDIX  I  (NOTE  TO  CHAPTER  XI,  §  13) 

52.  Price  and  Quantity  Movements  of  Rubber  and  Skins  and 
their  Average  by  Formulae  53  and  54 414 

53.  Price  and  Quantity  Movements  of  Lumber  and  Wool  and  their 
Average  by  Formulae  53  and  54 414 

54.  Index  Numbers  by  Formulae  353  and  2153  for  Rubber  and 
Skins 415 

55.  Index  Numbers  by  Formulae  353  and  2153  for  Lumber  and 
Wool 415 

APPENDIX  I  (NOTE  TO  CHAPTER  XIII,  §  9) 

56.  Cross  References  between  the  Numbers  for  Formulae  tabulated 
in  the  Purchasing  Power  of  Money  and  the  Numbering  used  in 
this  Book 419 

57.  Showing  the  Formulae  which  fulfill  and  do  not  fulfill  Three 
Supplementary  Tests 422 

APPENDIX  I  (NOTE  TO  CHAPTER  XV,  §  2) 

58.  Formula  2153P  as  a  Percentage  of  Formula  353P  (according  to , 

54 

Various  Values  of  —  and  353Q,  both  expressed  in  per  cents)  430 
53 

APPENDIX  II  (§  6) 

59.  Comparative  Effects  on  the  Index  Number  for    1917  (by 
Formula  3)  of  increasing  Tenfold  the  Weights  of  Half  of  the  36 
Commodities  according  as  the  Commodities  are  taken  at  Ran- 
dom, or  selected  to  make  the  Largest  Effect 446 


LIST  OF  TABLES  xxvii 

TABLE  PAGE 

60.  Index  Numbers  computed  by  using  Different  Weights  for 
Skins 446 

61.  Index  Numbers  computed  by  using  Different  Weights  for  Hay  447 

•  APPENDIX  V  (§3) 

62.  Formulae  for  Index  Numbers 466 

APPENDIX  VI  (§  6) 

63.  Prices  of  the  36  Commodities,  1913-1918 489 

64.  Quantities  Marketed  of  the  36  Commodities,  1913-1918      .  490 

APPENDIX  VII 

65.  Index  Numbers  by  134  Formulae  for  Prices  by  the  Fixed  Base 
System  and  (in  noteworthy  cases)  the  Chain  System      .      .  498 


LIST  OF  CHARTS 

(In  all  cases  the  charts  are  plotted  on  the  "ratio  chart"  in  which  the  vertical  scale  is  so 
arranged  that  the  same  slope  always  represents  the  same  percentage  rise.  For  a  full  descrip- 
tion of  the  advantages  of  this  method  the  reader  is  referred  to  "The  Ratio  Chart,"  Irving 
Fisher,  Quarterly  Publication  of  the  American  Statistical  Association,  June,  1917.) 

CHART  PAGE 

1.  Averaging  Two 5 

2.  Averaging  Three 6 

3.  Individual  Prices  and  Quantities  Dispersing  from  1913     .      .    12 

4.  Year  to  Year  Dispersion  of  Prices  and  Quantities       ...    20 

5.  Simple  Arithmetic  Index  Number  of  Prices  of  36  Commodities 
compared  with  "No.  353" 24 

6.  Index  No.  353  of  Prices  contrasted  (1)  with  dotted  lines  above, 
each  diverging  5%  in  a  year;  (2)  with  dotted  lines  below,  each 
diverging  1%  in  a  year 25 

7.  Uniform  Slopes  =  Uniform  Ratios 26 

8.  Simple  Index  Numbers  of  Prices  and  Quantities         ...    32 

9.  The  Five-Tine  Fork  of  6  Curves 50 

10.  The  Two  Extreme  Methods  of  Weighting  the  Median      .      .    52 

11.  Forward  ('17-' 18)  and  Backward  ('18-' 17)  Simple  Arithmetics 
contrasted 68 

12.  P(by  Formula  353)  x  Q(by  Formula  353)  =  V    178%  x  125% 

=  223% 76 

13.  P(by  Formula  9)  xQ(by  Formula  9)  not  =  to  V   187%  x  132% 
not  =  223% 77 

14.  Type  Bias  of  Formula  1 88 

15.  Three  Types  of  Index  Numbers:  Arithmetic,  Geometric,  Har- 
monic      92 

16.  Four  Methods  of  Weighting  compared:  By  base  year  values, 

by  mixed  values  (in  two  ways),  and  by  given  year  values    .      .    98 

17.  Four  Methods  of  Weighting  compared  for  Medians    .      .      .100 

18.  Double  Bias  (Weight  Bias  and  Type  Bias)  of  Formula  9    .      .102 

19.  Weight  Bias  of  Formula  29          104 

20.  The  Five-Tine  Fork  of  18  Curves 106 

21.  Insensitiveness  of  Median  and  Mode  to  Number  of  Commodi- 
ties .  114 


xxx  LIST  OF  CHARTS 

CHART  PAGE 

22.  The  Harmonic  Forward  is  Parallel  to  the  Arithmetic  Backward  120 

23.  Three  Types  of  Index  Numbers:  Factor  Antitheses  of  Har- 
monic, Geometric,  Arithmetic .      .  126 

24.  Four  Methods  of  Weighting  compared 128 

25.  The  Simple  Geometric :  compared  with  the  Simple  Arithmetic 
and  Harmonic  and  their  Rectification  by  Test  1         ...  138 

26.  Rectified  Arithmetic  and  Harmonic,  Simple         ....  146 

27.  Rectified  Geometric,  Simple 150 

28.  Rectified  Median,  Simple 152 

29.  Rectified  Mode,  Simple         .      .      ...      ......  154 

30.  Rectified  Aggregative,  Simple 156 

31.  Simple  Index  Numbers  and  their  Antitheses  and.  Derivatives: 
Satisfying   neither  Test;  satisfying  Test  1  only;  satisfying 
Test  2  only;  satisfying  both  Tests  (Modes  omitted)    .      .      .  158 

32.  Rectified  Arithmetic  and  Harmonic,  Weighted  (by  Values  in 
One  Year) 160 

33.  Rectified  Arithmetic  and  Harmonic,  Weighted  (by  "Mixed" 
Values)         162 

34.  Rectified  Geometric,  Weighted  (by  Values  in  One  Year)         .  164 

35.  Rectified  Geometric,  Weighted  (by  "Mixed"  Values)       .      .  166 

36.  Rectified  Median,  Weighted  (by  Values  in  One  Year)       .      .168 

37.  Rectified  Median,  Weighted  (by  "Mixed  "Values)     .      .      .170 

38.  Rectified  Mode,  Weighted 172 

39.  Rectified  Aggregative,  Weighted 174 

40.  Weighted  Index  Numbers  and  their  Antitheses  and  Deriva- 
tives: Satisfying  neither  Test;  satisfying  Test  1  only;  satisfying 
Test  2  only;  satisfying  both  Tests  (Modes  omitted)   .      .      .176 

41.  Range  of  Prices  and  Quantities  and  of  Three  Types  of  Index 
Numbers;  Weighted:  Satisfying  neither  Test;  satisfying  only  1 

or  only  2;  satisfying  both  1  and  2  (Modes  and  Medians  omitted)  178 

42.  Weighted  Index  Numbers  Doubly  Rectified  (Modes  omitted)  180 

43.  Close  Agreement  of  Cross  Formulae  and  Cross  Weight  Formulae  188 

44.  Close  Agreement  of  Cross  Formulae  and  Cross  Weight  Formulae 
(fully  rectified) 192 

45.  Weighted  Aggregatives  for  90  Raw  Materials:  War  Industries 
Board  Statistics 232 

46.  Formulae  53  and  54  applied  to  Stock  Market       .      .      .      .234 

47.  Formulae  53  and  54  applied  to  12  Leading  Crops  (after  W.  M. 
Persons),  1880-1920 ...  236 


LIST  OF  CHARTS  xxxi 

CHART  PAGE 

48.  Formulae  53  and  54  applied  to  12  Leading  Crops  (after  W.  M. 
Persons),  1910-1919 238 

49.  Ranking  as  to  Accuracy  of  Ail  Index  Numbers: 

1.  Worthless  Index  Numbers  249 

2.  Poor  Index  Numbers  250 

3.  Fair  Index  Numbers  252 

4.  Good  Index  Numbers  254 

5.  Very  Good,   (6)  Excellent,  and   (7)  Superlative  Index 
Numbers 255 

50.  Simple  Geometric  and  Simple  Median  compared  with  Ideal  for 
Different  Numbers  of  Commodities 263 

51.  Circular  Test:  Gaps  for  Years  0-4-5  of  Formulae  1,  9,  23,  141, 

151 279 

52.  Circular  Test:  Largest  Gaps  for  3-Around,  4-Around,  5- 
Around,  6-Around  Comparisons 286 

53.  Dispersion  (measured  by  Standard  Deviations)  (Fixed  Base) .  290 

54.  Dispersion  (measured  by  Standard  Deviations)  (Chain)  .      .  292 

55.  Dispersion  (measured  by  Standard  Deviations)  (Fixed  Base) 
(Sauerbeck's  Figures) 294 

56.  Comparison  for  Six  Bases  of  Formulae  53,  54,  353      ...  304 

57.  Optional  Varieties  of  353 .      .308 

58.  Simple  Median  and  Quartiles  drawn  from  Origin        .      .      .310 

59.  353  and  6023  compared  for  12  Leading  Crops,  1880-1919  (Day 
and  Persons) 314 

60.  353  and  6023  compared  for   12  Leading  Crops,  1910-1919 
(Day  and  Persons) 316 

61.  Effect  of  Number  of  Commodities  on  Index  Numbers        .      .  338 

62.  Finding  the  Simple  Mode 372 

63.  Different  Cross  Weightings  of  53  and  54 400 

64.  353  compared  with  its  Cross  Weight  Rivals          .      .      .      .404 

65.  Distribution  of  1437  Price  Relatives,  Forward  and  Backward  409 

66.  Simple  vs.  Cross  Weighted  Index  Numbers 440 

67.  Simple  vs.  Cross  Weighted  Index  Numbers,  Factor  Antitheses  442 

68.  Weighting  is  relatively  unimportant 448 


THE  MAKING 
OF  INDEX  NUMBERS 

CHAPTER  I 
INTRODUCTION 

§  1.   Objects  of  the  Book 

FOR  those  who  have  made  any  attempt  to  penetrate 
their  mysteries,  index  numbers  seem  to  have  a  perennial 
fascination.  Because  of  recent  upheavals  of  prices,  the 
interest  in  this  method  of  measuring  such  upheavals  is 
rapidly  spreading.  During  the  last  generation  index 
numbers  have  gradually  come  into  general  use  among 
economists,  statisticians,  and  business  men.  The  skepti- 
cism with  which  they  were  once  regarded  has  steadily 
diminished.  In  1896,  in  the  Economic  Journal,  the  Dutch 
economist,  N.  G.  Pierson,  after  pointing  out  some  ap- 
parently absurd  results  of  index  numbers,  said:  "The 
only  possible  conclusion  seems  to  be  that  all  attempts 
to  calculate  and  represent  average  movements  of  prices, 
either  by  index  numbers  or  otherwise,  ought  to  be  aban- 
doned." No  economist  would  today  express  such  an 
extreme  view.  And  yet  there  lingers  a  doubt  as  to  the 
accuracy  and  reliability  of  index  numbers  as  a  means  of 
measuring  price  movements. 

It  is  perfectly  true  that  different  formulae  for  calculat- 
ing index  numbers  do  yield  different  results.  But  the 
important  question,  never  hitherto  answered  in  a  com- 
prehensive way,  is:  How  different  are  the  results,  and 

l 


2        ;TKE  -MAKING  OF  INDEX  NUMBERS 

can  we  find  reasons  for  accepting  some  and  rejecting 
others? 

To  answer  this  general  question  as  to  the  trustworthi- 
ness of  index  numbers  is  one  of  the  two  chief  purposes  of 
the  present  book.  In  order  to  make  the  answer  conclu- 
sive, all  the  formulae  for  index  numbers  which  have  been 
or  could  reasonably  be  constructed,  have  been  investi- 
gated and  tested  in  actual  calculations  based  on  actual 
statistical  records.  We  shall  find  that  some  of  the  for- 
mula in  general  use  and  unhesitatingly  accepted  by  un- 
critical users  are  really  very  inaccurate,  while  others  have 
an  extraordinary  degree  of  precision.  The  reasons  for 
these  differences  will  be  investigated  as  well  as  the  attri- 
butes essential  to  precision. 

The  second  chief  purpose  of  this  book  is  to  help  make 
the  calculation  of  index  numbers  rapid  and  easy.  To  this 
end  we  shall  show  what  formulae  are  best  in  theory  and 
practice,  and  shall  indicate  certain  short  cuts  for  their 
calculation. 

§  2.  An  Index  Number  Defined 

Most  people  have  at  least  a  rudimentary  idea  of  a  "  high 
cost  of  living"  or  of  a  "low  level  of  prices,"  but  usually 
very  little  idea  of  how  the  height  of  the  high  cost  or  the 
lowness  of  the  low  level  is  to  be  measured.  It  is  to  meas- 
ure such  magnitudes  that  "index  numbers"  were  in- 
vented. 

There  would  be  no  difficulty  in  such  measurement, 
and  hence  no  need  of  index  numbers,  if  all  prices  moved 
up  in  perfect  unison  or  down  in  perfect  unison.  But 
since,  in  actual  fact,  the  prices  of  different  articles  move 
very  differently,  we  must  emgloy^some  sort  of  compro- 
mise  or  average  of 


look  at  prices  as  starting  at  any  time  from  the 


INTRODUCTION  3 

same  point,  they  seem  to  scatter  or  disperse  like  the 
fragments  of  a  bursting  shell.  But,  just  as  there  is 
a  definite  center  of  gravity  of  the  shell  fragments, 
as  they  move,  so  is  there  a  definite  average  move- 
ment of  the  scattering  prices.  This  average  is  the  "  index 
number."  Moreover,  just  as  the  center  of  gravity  is 
often  convenient  to  use  in  physics  instead  of  a  list  of  the 
individual  shell  fragments,  so  the  average  of  the  price 
movements,  called  their  index  number,  is  often  convenient 
to  use  in  economics. 

^An  index  number  of  prices,  then,  shows  the  average  L 
percentage  change  of  prices  from  one  point  of  time  to  an-  [|j! 
other.     The  percentage  change  in  the  price  of  a  single 
commodity  from  one  time  to  another  is,  of  course,  found 
by  dividing  ij£  price  at  the  second  time  by  its  price  at 
the  first  tirr^e.     The  ratio  between  these  two  prices  is 
called  the  price  relative  of  that  one  particular  commodity 
in  relation  to  those  two  particular  times.    An  index  num- 
ber of  the  prices  of  a  number  of  commodities  is  an  average 
of  their  price  relatives. 

This  definition  has,  for  concreteness,  been  expressed 
in  terms  of  prices.  But  in  like  manner,  an  index  number 
can  be  calculated  for  wages,  for  quantities  of  goods  im- 
ported or  exported,  and,  in  fact,  for  any  subject  matter 
involving  divergent  changes  of  a  group  of  magnitudes. 

Again,  this  definition  has  been  expressed  in  terms  of 
time.  But  an  index  number  can  be  applied  with  equal 
propriety  to  comparisons  between  two  places  or,  in  fact, 
to  comparisons  between  the  magnitudes  of  a  group  of 
elements  under  any  one  set  of  circumstances  and  their 
magnitudes  under  another  set  of  circumstances.  But 
in  the  great  majority  of  cases  index  numbers  are  actually 
used  to  indicate  mice  rnomments  in  time. 


4  THE  MAKING  OF  INDEX  NUMBERS 

§  3.  Illustrations  —  Numerical,  Graphic,  Algebraic 

An  index  number  is  an  average.  There  are  many  kinds 
of  averages  —  the  arithmetic,  the  geometric,  etc.,  of  which 
only  the  arithmetic  is  known  to  most  people.  In  these 
preliminary  illustrations,  therefore,  we  shall  employ  the 
arithmetic  average,  but  always  specify  " arithmetic"  in 
order  not  to  lose  sight  of  the  fact  that  this  is  but  one  kind 
of  average. 

Numerically,  if  wheat  has  risen  4  per  cent  since  some 
specified  date,  say  January  1,  1920  (say  from  $1.  a 
bushel  to  $1.04),  and  beef  has  risen  10  per  cent  in  the  same 
time  (say  from  10  cents  per  pound  to  11),  the  simple 
arithmetic  average  percentage  rise  of  wheat  and  beef  is 
midway  between  4*  per  cent  and  10  per  cent,  or  7  per( 

4-4-10 

cent  (that  is,    T       ,7).    Then  107  per  cent  is  the  "in- 
2i 

dex  number"  for  the  present  prices  of  these  two  articles- 
as  compared  with  those  of  the  original  date,  called  the 
"base"  and  taken,  for  convenience,  as  100  per  cent.  Or : 


COMMODITY 

JANUAKT  1,  1920 

PRESENT  TIME 

Wheat  

100  per  cent 

104  per  cent 

Beef  

100  per  cent 

110  per  cent 

Simple  arithmetic  average  

100  per  cent 

107  per  cent 

Thus  107  per  cent  is  an  index  number  based  on  the  twc 
price  ratios,  or  "price  relatives,"  104  per  cent  and  110 
per  cent. 

Graphically,  Chart  1  pictures  the  numerical  results  given 
above. 

Algebraically,  if  the  price  of  one  commodity  in  1920 
(January  1)  is  p0  and,  in  1921,  pi,  and  the  price  of  another 
commodity  in  1920  is  p'Q  and,  in  1921,  p\,  then  their 


INTRODUCTION 

Averaging     Two 


1911 


CHAKT  1.  Percentage  changes  in  price  of  two  commodities  and  the  average 
percentage  change. 

price  ratios  or  "price  relatives"  are  —  and  ^>  and  the 

Po          p  o 

simple  arithmetic  average  of  the  two,  that  is,  the  simple 


arithmetic  index  number,  is    °         °.  It  is  convenient  to 

2i 

multiply  the  result  by  100  to  express  it  in  percentages. 

The  same  method  applies,  of  course,  to  more  than  two 
prices.  Thus,  if  three  prices,  say  sugar,  wheat,  and  beef, 
rise  respectively  4  per  cent,  4  per  cent,  and  10  per  cent, 

their  average  rise  is  '  —  or  6  per  cent,  and  the  "in- 

o 

dex  number"  is  106  as  compared  with  the  original  price 
level  of  100  taken  as  a  base  of  comparison. 

Graphically,  Chart  2  shows  the  simple  arithmetic  aver- 
age just  described. 

Algebraically,  the  simple  arithmetic  index  number  of 
three  commodities  is  evidently 


Po  _  p  o      P    o 


6  THE  MAKING  OF  INDEX  NUMBERS 

Averaging    Three 

of  oil  *«* 


/OO^ 

Jan.  /  i9zo  Jan  / /a?/ 

CHART  2.   Percentage  changes  in  price  of  three  commodities  and  the 
average  percentage  change. 

§  4.  Weighting 

The  preceding  calculation  treats  all  the  commodities 
as  equally  important ;  consequently,  the  average  was 
called  "simple."  If  one  commodity. is  more  important 
than  another,  we  may  treat  the  more  important  as  though 
it  were  two  or  three  commodities,  thus  giving  it  two  or 
three  times  as  much  "weight"  as  the  other  commodity. 

Thus,  suppose  that  wheat  is  taken  to  be  twice  as  im- 
portant as  beef.  Then  the  average  rise  of  wheat  and 

beef,  instead  of  being  — t —  =  7^  as  it  was  when  the  two 
commodities  were  regarded  as  equally  important,  becomes 

>        Q     —  =  6,  just  as  though  there  were  three  commodi- 
o 

ties,  thus  making  the  index  number  106  instead  of  107. 
In  this  average,  wheat  is  weighted  twice  as  heavily  as 
beef.  If,  reversely,  beef  is  given  twice  as  much  weight 
in  determining  the  index  number  as  wheat,  the  average 

rise  is  4 +(10 +  10)  =,§  an(j  the  index  number  is  108  in- 
o 

stead  of  107.  w 


INTRODUCTION  7 

Algebraically,  if  the  wheat  is  weighted  twice  as  heavily 
.as  the  beef  —  that  is,  if  their  weights  are  as  2  to  1  —  the 
formula  for  this  weighted  arithmetic  index  number  be- 
^  comes 


3 

It  makes  no  difference  to  the  result  whether  the  weights 
be  2  and  1  as  above,  or  4  and  2,  or  20  and  10,  or  any  other 
two  numbers  of  which  one  is  double  the  other,  since  the 
denominator  increases  proportionally.  Thus,  if  the 
weights  were  14  and  7  the  formula  would  be 


14(2!) 

V/ 


21 

which  could  evidently  be  reduced  to  the  first  formula 
simply  by  canceling  "7"  in  the  numerator  and  denomi- 
nator. 

Thus  "weighting"  is  clearly  relative  only.  If  we  weight 
wheat  and  beef  evenly,  say  10  and  10,  evidently  the  re- 
sult is  the  simple  average.  So  a  simple  average  may  be 
said  to  be  a  weighted  average  in  which  the  weights  are  all 
equal.  Strictlyspeaking,  therefore,  there  is  no  such  thing 
asan  imweigEted  average.^ 

In  general  algebraic  terms,  if  the  weight  for  wheat  is 
w  and  that  for  beef  is  wf,  the  weighted  arithmetic  average 

(£)+-•(£) 

W  +  W' 

Graphically,  the  effect  of  weighting  wheat  heavily  is 
evidently  to  bring  the  index  number  line  of  Chart  1  down 
nearer  to  the  wheat  line  as  in  Chart  2,  while  weighting 
beef  more  heavily  swings  it  up  toward  the  beef  line. 


- 


8  THE  MAKING  OF  INDEX  NUMBERS 

We  have  illustrated  the  two  most  common  varieties  of 
index  numbers,  the  simple  arithmetic  and  the  weighted 
arithmetic,  or,  as  they  might  in  strict  accuracy  be  called, 
the  evenly  weighted  and  the  unevenly  weighted  arith- 
metic index  numbers.  But,  as  already  noted,  there  are 
ymany  kinds  of  index  -number  formulae  other  than  the 
arithmetic.  In  fact,  there  are  as  many  possible  varieties 
of  formulae  as  there  are  different  varieties  of  averages, 
and  these  are  infinite. 

§  6.  Attributes  of  an  Index  Number 

Moreover,  index  numbers  differ  from  each  other  not 
only  as  to  the  kinds  of  formulae  used  in  calculating  them, 
but  also  in  several  other  respects,  or  "attributes." 
Briefly,  all  the  attributes  of  an  index  number,  twelve  in 
number,  may  be  enumerated  under  three  groups  as  fol- 
lows: 

I.  As  TO  THE  CONSTRUCTION  OP  THE  INDEX  NUMBER 

(1)  The  general  character  of  the  data  included,  e.g. ' '  whole- 
sale prices "  or  "retail  prices "  of  commodities,  or  "prices 
of  stocks,"  or  "wages,"  or  "volume  of  production,"  etc. 

(2)  The  specific  character  of  data  included,  e.g.  "foods," 
still  further  specified  as  "butter,"    "beef,"  etc. 

(3)  Their  assortment,  e.g.  a  larger  proportion  of  quo- 
tations of  meats  than  of  vegetables. 

(4)  The  number  of  quotations  used,  e.g.  "22  commodi- 
ties" as  in  the  case  of  the  Economist  index  number  (until 
recently)  as  contrasted  with  "1474  commodities"  as  in 
the  case  of  the  War  Industries  Board. 

j  (5)  The  kind  of  mathematical  formula  employed  for 
calculating  the  index  number,  e.g.  the  "simple  arithmetic 
average"  or  the  "weighted  geometric  average,"  etc. 


INTRODUCTION  9 

II.   As  TO  THE  PARTICULAR  TIMES  OR  PLACES  TO  WHICH 
THE  INDEX  NUMBER  APPLIES 

(1)  The  period  covered,  e.g.  "1913-1918,"  or  the  terri- 
tory covered,  e.g.  certain  specified  cities  of  which  the 
price  levels  are  to  be  compared. 

(2)  The  base,  e.g.  the  year  1913. 

(3)  The  interval  between  successive  indexes,  e.g.  "yearly  " 
or  "  monthly." 

III.   As  TO  THE  SOURCES  AND  AUTHORITIES 

(1)  The  agency  which  collects,  calculates,  and  publishes 
the  index  number,  e.g.   "  Brads treet's"  or  the  "  United 
States  Bureau  of  Labor  Statistics." 

(2)  The  markets  used,  e.g.  the  "Stock"  or  "Produce" 
Exchanges  of  "New  York"  or  the  "primary  markets  of 
the  United  States." 

(3)  The  sources  of  quotations,  e.g.   the  "leading   trade 
journals"  or  the  books  of  business  houses. 

(4)  The  publication  containing  the  index  number,   e.g. 
the  Bulletin  of  the  United  States  Bureau  of  Labor  Sta- 
tistics. 

Of  these  12  attributes  characterizing  an  index  number, 
I  shall  deal  in  detail  with  one  only,  namely,  the  formula. 
The  other  11  attributes,  previous  writers  have  covered 
to  a  large  extent,  and  I  shall  content  myself  with  a  very 
brief  summary  of  their  conclusions,  which  will  be  given 
at  the  end  of  this  book. 

§  6.  Fairness  of  Index  Numbers 

The  multiplicity  of  formulae  for  computing  index  num- 
bers has  given  the  impression  that  there  must  be  a  corre- 
sponding multiplicity  in  the  results  of  these  computations, 
with  no  clear  choice  between  them.  But  this  impression 


10          THE  MAKING  OF  INDEX  NUMBERS 

is  due  to  a  failure  to  discriminate  between  index  numbers 
which  are  good,  bad,  and  indifferent.  By  means  of  cer- 
tain tests  we  can  make  this  discrimination. 

The  most  important  tests  are  all  embraced  under  the 
single  head  of  fairness.  The  fundamental  purpose  of  an 
index  number  is  that  it  shall  fairly  represent,  so  far  as  one 
single  figure  can,  the  general  trend  of  the  many  diverging 
ratios  from  which  it  is  calculated.  It  should  be  the  "  just 
compromise"  among  conflicting  elements,  the  "fair  aver- 
age," the  "  golden  mean."  Without  some  kind  of  fair 
splitting  of  the  differences  involved,  an  index  number  is 
apt  to  be  unsatisfactory,  if  not  absurd.  How  we  are  to 
yfcest  the  fairness  of  an  average  will  be  shown  in  Chapter 
IV. 

Meanwhile  it  will  be  advisable  first,  to  describe  the 
various  types  of  index  numbers  ;  for,  thus  far,  we  have  dis- 
cussed only  the  arithmetic  type. 


CHAPTER  II 
SIX  TYPES  OF  INDEX  NUMBERS  COMPARED 

§  1.  The  Dispersion  of  Individual  Prices  and  Quantities 

As  a  preliminary  to  calculating  various  kinds  of  index 
numbers  we  may  picture  the  movements  of  the  36  in- 
dividual commodities  which  will  be  used  for  the  compari- 
sons in  this  book. 

Graphically,  Chart1  3P  shows  the  movements  of  the 
prices  of  these  36  commodities  considered  as  diverging 
from  a  common  starting-point  in  1913,  and  Chart  3Q 
shows,  in  like  manner,  the  movements  of  the  quantities 
marketed  of  these  same  36  commodities. 

A  casual  observer,  looking  at  the  diverging  and  tangled 
course  of  prices  and  quantities,  would  be  tempted  to  give 
up  in  advance,  not  only  any  attempt  to  find  index  numbers 
which  can  truly  represent  changes  in  the  " general  trend" 
of  these  widely  scattering  figures,  but  also  to  wonder 
whether  the  words  " general  trend"  corresponded  to  any 
real  and  clear  idea.     He  would  note  that  at  the  close  of 
the  period,  in  1918,  the  price  of  rubber  was  32  percent 
below  its  starting-point,  in  1913,  while  the  price  of  wool 
was  182  per  cent  above  its  starting-point.     Thus,  their 
price  relatives,  in  1918  relatively  to  1913,  are  as  68.02  to 
^  100  and  as  282.17  to  100,  the  latter  being  4  times  the 
"  former,  with  the  other  34  price  relatives  widely  scattered 
-  between.     As  to  quantities,  he  would  find  that  the  quan- 
tity of  rubber  in  1918  stood  at  303.54  and  that  of  skins 

1  All  charts  in  this  book  are  "ratio  charts,"  as  explained  in  detail  later 
in  this  chapter. 

11 


12 


THE  MAKING  OF  INDEX  NUMBERS 


Individual 
Prices 

Dispersing    from 
/9I3 


petroleum 
lumber 
anth.  coal 
hides 


coffee 


rubber 


14 


16 


17 


'18 


CHART  3P.  Showing  the  enormously  wide  dispersion  of  the  price  move- 
ments of  the  36  commodities.     (The  eye  is  enabled  to  judge  the  relative 
vertical*  positions  df  the  curves  in  this  and  other  charts  by  means  of  the 
littjETdark  vertical  line  marked  "5  %"  inserted  to  serve  as  a  measuring  rod. 
^  Thus  in  1917  coffee  is  about  5  per  cent  higher  than  rubber  while  petroleum 
v    is  about  JO  per  cent  higher  than  coffee  and  anthracite  coal  10  per  cent 
higher  than  petroleum.) 


SIX  TYPES  OF  INDEX  NUMBERS  13 

•rubber 


Individual 
Quantities 

Dispersing    from 


slumber 
Jime 

cement 
white  lead 

mutton 
steel 


skins 


75         14          75          J6          17          /8 

CHART  3Q.  Showing  the  enormously  wide  dispersion  of  movements  of 
the  quantities  marketed  of  the  36  commodities. 


14  THE  MAKING  OF  INDEX  NUMBERS 

at  10.45  (too  low  to  get  on  the  chart)  so  that  the  former 
was  29  times  the  latter,  with  the  other  34  quantity  rela- 
tives widely  scattered  between. 

How  is  it  possible  to  find  a  common  trend  for  such 
widely  scattered  price  relatives  or  quantity  relatives? 
Will  not  there  be  as  many  answers  to  such  a  question  as 
there  are  methods  of  calculation  ?  Will  not  these  answers 
vary  among  themselves  50  per  cent  or  100  per  cent?  The 
present  investigation  will  show  how  mistaken  is  such  a 
first  impression. 

§  2.  Uniform  Data  Used  for  Comparisons 

The  36  price  movements  and  the  36  quantity  move- 
ments just  pictured  will  constitute  the  raw  material  for 
calculating  the  many  kinds  of  index  numbers  which  we 
shall  consider.  Thus  the  very  same  data  will  be  used 
for  calculating  different  kinds  of  index  numbers  by  134 
different  formulae.  These  data  are  a  part  of  the  mass  of 
statistics,  collected  by  Wesley  C.  Mitchell  for  the  War 
Industries  Board,  for  wholesale  prices  and  quantities 
marketed  of  1474  commodities  in  the  United  States.  The 
list  of  these  36  commodities  and  the  figures  for  the  prices 
and  quantities  of  each  are  given  in  Appendix  VI,  §  1. 

One  chief  reason  for  employing  data  from  the  records 
of  the  War  Industries  Board  is  that  they  are  based  on  the 
only  1  collection  of  data  which  includes  figures  for  quanti- 
ties as  well  as  for  the  prices  of  each  commodity.  This 
same  set  of  data  is  used  for  all  of  the  comparisons  under 
the  various  formulae.  We  may  be  sure  that  our  tests 
are  severe  and  conclusive  because  the  period  covered, 
1913  to  1918,  is  (as  will  be  shown  statistically,  later)  a 
period  of  extraordinary  dispersion  in  the  movements 
both  of  prices  and  quantities. 

1  Since  the  present  work  was  begun  there  have  appeared  the  studies 
by  Professors  Day  and  Persons  of  12  commodities  cited  later. 


SIX  TYPES  OF  INDEX  NUMBERS  15 

*  In  view  of  this  fact  we  may  be  confident  that  the  close- 
ipess  of  agreement,  which  the  following  calculations  show 
II  among  those  index  numbers  which  are  not  demonstrably 

f  I  unfair  in   their   construction,  does  not   exaggerate  but 
/  actually  understates  the  closeness  which  will  be  encoun- 
||  tered  in  ordinary  practice. 
t 

§  3.  The  Simple  Arithmetic  Average  of  Relative  Prices 
by  the  Fixed  Base  System 

•  Although  we  shall  calculate  index  numbers  by  13^ 
.  different  formulae,  they  all  fall  under  six  types  :  the  arith- 
metic, harmonic,  geometric,  median,  mode,  and  aggrega-j| 
tive.1    These  are  the  only  types  of  average  ever  considered 
for  index  numbers,  or  ever  likely  to  be  considered,  and  one 
of  them,  the  mode,  might  almost  have  been  omitted  as 
never  having  been  seriously  proposed  for  actual  use,  al- 
though often  referred  to  in  connection  with  the  subject. 

None  of  the  six,  except  the  simple  arithmetic  average 
^of  relative  prices,  are  familiar  to  most  people.     In  fact 
'  the  very  word  "  average "  means,  to  most  people,  only  the 
^simple  arithmetic  average.     Let  us,  therefore,  begin  by  > 
^defining  this  kind  of  average  in  order  to  differentiate  it 
others, 
simple  arithmetic  average  of  a  number  of  terms 

their  sum  divided  by  the  number  of  the  terms.  Thus 
^  to  average  3  and  4  we  divide  their  sum.  (7)  by  their  num- 
'  ber  (2)  and  obtain  3i  as  the  simple  arithmetic  average  of 
£3  and  4.  Again,  averaging  likewise  5,  6,  and  7  we  get 

^5+ o  +  7  =  6>   and   averaging   8,   8.5,   9,   9.7  we   get 

^8  -F  8.5  +  9  +  9.7  =8>g< 
4 

*As  to  the  word  "aggregative"  see  Appendix  I  (Note  A  to  Chapter 
II,  §  3). 


16 


THE  MAKING  OF  INDEX  NUMBERS 


To  apply  this  sort  of  calculation  to  index  numbers, 
let  us  take  the  following  skeleton  table  of  the  prices  of  our 
36  commodities  for  the  two  years,  1913  and  1914 : 

TABLE  1.  THE  SIMPLE  ARITHMETIC  INDEX  NUMBER 
FOR  1914  AS  CALCULATED  FROM  THE  36  PRICES  FOR 
1913  AND  1914 


No. 

COMMODITY 

PRICES 
IN  CENTS 

PRICE 
RELATIVES 

1913 

1914 

wo  x  if 

1 
2 

"12.36 
-62.63 

12.95 
-£2.04 

^104.77 
^99.06 

Barley,  per  bu  

36 

Oats,  per  bu  

37.58 

41.91 

^111.52 

36 

)  3467.36 
96.32 

he  first  two  columns  of  figures  give  the  actual  prices, 
the  last  column  gives  the  relative  prices,  found  by  calling 
each  price  in  1913  100  per  cent,  while  the  average  of  these 
is  the  index  number  sought. 

Thus,  to  obtain  the  index  number  of  these  commodities 
for  1914,  relatively  to  1913  as  the  base,  two  steps  are  in- 
volved :  first,  to  get  the  relation  between  each  commodi- 
ty's 1914  price  and  its  jj)13,  or  base,  price.  This  is  a 
ratio.  It  is  expressed  in  percentages  and  is  called  a  rela- 
tive price  or  "  price  relative. "  There  is,  thus,  a  price 
relative  for  bacon,  another  price  relative  for  barley,  and 
so  on  —  a  price  relative  for  each  separate  commodity. 
To  obtain  these  price  relatives  is  the  first  step  to  an  index 
number  and  may  be  called  "percentaging."  The  second 
step  is  to  average  these  relatives  —  and  may  be  called 
"  averaging  the  percentages." 

The  first  item  on  the  list  is  bacon,  the  price  of  which 


SIX  TYPES  OF  INDEX  NUMBERS  17 

in  1913  was  12.36  cents  per  pound  and,  in  1914,  12.95 
'cents  per  pound,  which  is  4.77  per  cent  higher.  That  is, 
percentaging  the  prices  of  bacon  we  find  the  price  in  1914, 
relatively  to  1913,  to  be  100  X  (12.95  ^  12.36),  or  104.77 
per  cent.  Likewise,  barley  fell  from  62.63  cents  per 
bushel  to  62.04,  the  latter  being  99.06  per  cent  of  the 
former.  Thus,  99.06  per  cent  is  the  price  relative  of  bar- 
ley (for  1914  relatively  to  1913  taken  as  100),  and  so  on 
to  the  end,  where  oats  rose  in  1914  to  111.52,  as  compared 
with  100  taken  as  its  price  in  1913. 

Having  thus  percentaged  the  prices  into  price  relatives, 
we  proceed  to  average  the  percentages.  The  simple 
arithmetic  average  of  these  price  relatives,  namely,  of 
104.77,  99.06,  .  .  .  ,  111.52,  is  found  by  first  taking  their 
sum  (3467.36)  and  then  dividing  this  sum  by  their 
number  (36).  The  result  is  96.32  per  cent,  the  desired 
simple  arithmetical  index  number,  giving  the  price  level 
of  1914  as  a  percentage  relatively  to  100  in  1913  as  the 
base  of  comparison.  The  base  is  the  year  for  which  each 
price  is  taken  as  100  per  cent  (or  any  other  common 
figure).1 

In  the  same  way,  the  simple  arithmetic  index  number 
for  1915  relatively  to  100  in  1913 -as  a  base  is  98.03,  or 
L97  per  cent  below  1913 ;  and  for  the  next  three  years, 
1916,  1917,  and  1918  respectively,  the  simple  arithmetic 
index  numbers  are  123.68,  175.79,  186.70  —  all  relatively 
to  100  in  1913  as  base  —  or  higher  than  1913  by  23.68 
per  cent,  75.79  per  cent,  and  86.70  per  cent  respectively. 
*  Sometimes  it  is  convenient  to  make  some  other  year 
than  the  base  JLOO  Aper  cent.  Thus,  we  might  wish  to 
translate  the  above  series  (100.00,  96.32,  98.03,  123.68, 
175.79,  186.70,  all  calculated  on  1913  as  a  base)  into  pro- 
portional numbers  with  100  in  place  of  186.70  for  1918. 
1  See  Appendix  I  (Note  B  to  Chapter  II,  §  3). 


18  THE  MAKING  OF  INDEX  NUMBERS 

The  series  then  becomes  53.56,  51.59,  52.51,  66.25,  94.16, 
100.00. 
•     But  this  replacement  of  the  awkward  number  186.70 

^in  1918  by  the  more  convenient  number  100,  and  the 
proportionate  reduction  of  the  original  100  in  1913  to 
53.56,  does  not  really  change  the  base  from  1913  to  1918. 
1913  is  still  the  base,  but  the  base  number  is  changed  from 
100  to jSJifi ;  for  the  base  number  is  the  number  common 

*  to  all  the  commodities.  Evidently  to  change  an  index 
number  for  £1918  from  186.70  to  100  does  not  make 
each  separate  commodity  100.  The  commodities  having 
before  had  36  different  numbers,  the  average  of  which 
was  186.70  will  now  have  36  different  numbers,  the  aver- 
age of  which  is  100.  On  the  other  hand,  1913,  which  be- 
fore had  every  commodity  100,  will  now  have  every 
commodity  53.56  ;  therefore,  1913  is  still  the  base.  Thus, 
we  must  sometimes  distinguish  between  the  true  base  year 
and  the  year  for  which  the  index  number  is  taken  as  100. 
After  a  series  of  index  numbers  has  been  computed  it 
is  very  easy  so  to  reduce  or  magnify  all  the  figures  in 
proportion,  or  to  make  any  year  lit  which  we  chose. 

§4.  The  Simple  Arithmetic  Average  of  Relative  Prices 
by  the  " Chain"  feystem 

In  the  preceding  discussion  all  the  index  numbers  were 
calculated  relatively  to  1913  as  a  common  base.  The 
price  of  every  one  of  the  36  commodities  was  taken  as 
100  per  cent  in  1913,  and  then,  by  percentaging,  the  price 
relatives  of  the  other  year  were  found,  and  then  averaged. 
But,  of  course,  any  other  year  could  be  used  as  the  base. 
Thus  we  might  take  1918  as  the  base  and  calculate  any 
other  year  relatively  to  Ifllg.  Or  we  could  use  one  base 
for  one  comparison  and  another  base  for  another  com- 
parison. If  every  one  of  our  six  years  were  used  as  the 


SIX  TYPES  OF  INP^^NUMBERS  19 

base  for  every  other  year,  we  would  have  30  index  num- 
bers in  all,  and  these  would  all  be  discordant  among 
themselves. 

The  usual  practice  is  to  keep  to  one  year  or  period — 
the  earliest  year  of  the  series,  or  sometimes  an  average  of 
several  years  —  as  the  base  for  the  calculation  of  the 
price  relatives.  This  "  fixed  base  "  method  gives  us  a  series 
of  figures  which,  in  practice7»ce  used  not  only  for  compar- 
ing each  year  with  1913,  but  forXcomparing  each  year  with 
the  one  before  or  after.  Thus,J#ie  last  two  figures,  175.79 
and  186.70,  are  regarded  as  showing  the  price  levels  of 
1917  and  1918  relatively  not  only  to  1913,  but  to  each 
other.  But  properly  to  measure  the  price  movement  be- 
tween the  two  years  1917  and  1918,  we  ought  not  to  be 
obliged  to  take  some  third  year,  like  1913,  as  a  base.  We 
should  be  able  to  compare  1917  and  1918  directly  with 
each  other.  By  the  "chajajof  bagoa_s3[stgm"  each  year1 
is  taken  as  the  base  fo'r  calculating  the^ndex  number 
of  the  next,  and  the  resulting  figures  are  then  linked  to- 
gether to  form  a  " chain"  of  figures.  This  will  be  cl£ar 
if  we  take  one  link  at  a  time. 

First,  we  calculate  the  index  number  of  1914  relatively 
to  IplSjts  a  base.  In  this  case  the  calculation  is  identical 
with  that  of  the  fixed  base  system  when  1913  is  the  base. 
We  have,  then,  the  first  link,  which  is  96.32  per  cent. 
Next,  we  calculate  the  index  number  of  1915  relatively, 
not  to  1913,  but  to  1914  as  the  base.  That  is,  we  per- 
centage the  prices  of  19JJLby  taking  each  price  of  1914  as 
100  per  cent,  thus  obtaining  36  price  relatives  quite  dif- 
ferent from  any  previously  calculated  under  the  fixed 
(1913)  ba'se  system;  and  then  average  these  36  price 
relatives.,  We  now  have  the  second  link.  This  is  101.69 
per  cent,  the  index  number  of  1915  relatively  to  1914  as 
100  per  cent. 


20  THE  MAKING  OF  INDEX  NUMBERS 

But  this  index  (of  1915  to  1914)  is  only  a  link  in  the 
chain.    We  must  still  join  it  to  the  preceding  link  to  ob- 


Yeor   to  Year  Dispersion 
of  Prices 


\5% 


73  '14  '15  '16  17  18 

CHART  4P.  The  lines  from  1913  to  1914  are  the  same  as  in  Chart  3P; 
the  lines  for  subsequent  years  are  parallel  to  their  positions  in  Chart  3P, 
but  are  shifted  so  as  to  start  over  again  from  a  new  common  point  in  each 
successive  year. 

tain  the  index  of  Jj)15  to  1913^^1914.  This  requires 
a  third  step,  namely,  multiplying  this  second  link  (1915 
to  1914)  by  the  first  (1914  to  1913),  thus :  lOJj&Q-per  cent  X 
96.32  per  cent  =  97.94l:)er  cent. 


SIX  TYPES  OF  INDEX  NUMBERS 


21 


In  the  same  way  we  calculate  the  third  link,  the  index 
number  for  1916  relatively  to  1915  as  a  base  (that  is,  by 
percentaging  relatively  to  1915  and  averaging  the  re- 

Year  fo  Year  Dispersion 
of    Quantities 


CHART  4Q.  Analogous  to  Chart  4P. 

suiting  price  relatives).  We  then  join  this  third  link 
(127.97  per  cent)  on  to  the  chain  by  multiplying  it  by  the 
two  previous  (127.97  per  cent  X  101.69  per  cent  X  96.32 
per  cent),  obtaining  125.33  per  cent  as  the  chain  figure 
for  1916  relative,  indirectly,  to  1913.  That  is,  this  is  the 
index  number  for  1916  relative  to  1913  as  100  per  cent, 
but  via  the  intermediate  bases,  1914  and  1915. 


! 


22  THE  MAKING  OF  INDEX  NUMBERS 

In  short,  by  this  chain  system,  or  step  by  step  system, 
each  year's  index  number  is  first  calculated  as  a  separate 
link  relatively  to  the  preceding  year  as  the  base.  But 
after^  these  separate  year-to-year,  or  link  index  numbers, 
are  thus__calculated  by  the  usual  two  processes  of  per- 
centaging  and  averaging,  they  are  joined  together  by  the 

or  successive  multiplications  to 


form  "  chain"  figures.  Consequently,  for  the  final  series 
only  the  initial  base,  1913,  stays  at  100  per  cent.  This 
third  process,  linking,  is  added  because  it  is  much  more 
convenient  to  have  only  one  100  per  cent  year  in  the  final 
series  than  to  use  the  year-to-year  links  in  which  each 
year  is  100  per  cent  for  the  next. 

•s. 

§  5.   Charts  Illustrating  the  Chain  System 

Graphically,  the  averaging  of  the  separate  links  is 
shown  in  Charts  4P  and  4Q,  where  the  prices  and  quan- 
tities are  pictured  as  dispersing,  first,  from  1913  to  the 
next  year,  and  then  from  1914  to  the  next,  and  so  on  by 
successive  steps.  Each  new  point  of  departure  is  taken 
as  an  average  of  the  preceding  set  of  lines  so  that  all  these 
points  constitute  the  chain  series  of  index  numbers. 

The  two  methods,  fixed  base  and  chain,  may,  of  course, 
be  applied  to  every  formula.  For  some  formulae  the  two 
methods  give  identical  results;  for  others,  not.  In  the 
case  of  the  simple  arithmetic  index  number  they  do  not. 

§  6.   The  Simple  Arithmetic,  Both  Fixed  Base  and  Chain. 
Illustrated  Numerically  and  Graphically 

Table  2  shows  the  simple  arithmetic  index  numbers  by 
both  methods  —  fixed  base  method  and  chain  method  — 
together  with  the  individual  links  of  the  chain.1 

1  Appendix  I  (Note  B  to  Chapter  II,  §  3)  might  profitably  be  consulted 
here. 


SIX  TYPES  OF  INDEX  NUMBERS 


23 


TABLE  2.    SIMPLE  ARITHMETIC  INDEX  NUMBER 
(FORMULA   I)1  FOR  PRICES 

(By  fixed  base  method  and  by  chain  method) 


1913 

1914 

1915 

1916 

1917 

1918 

1913  as  fixed  base 

100. 

«>6.32> 

^  "&8.03 

123.68 

175.79 

186.70 

1914  as  base  for  1915 
1915  as  base  for  1916 
1916  as  base  for  1917 

400.00 

*LOO.OO 

"127.  97N 
400.00 

^40.15 

1917  as  base  for  1918 

100.00 

•^lO.ll 

By  chain  of  above  bases 
(product  2  of  above 
links  successively) 

100. 

g6.32 

97.94 

125.33 

175.65 

193.42 

1  Complete  tables  of  the  index  numbers  reckoned  by  all  of  the  134  formulas  are  given 
in  Appendix  VII.  The  formulae  themselves  are  given  in  Appendix  V. 

*  97.94  is  obtained  by  multiplying  96^32  X  ^)1.69 ;  125.33  by  multiplying  96.32  X  101.69 
X  127.97,  etc.  In  multiplying,  we  must  remember  that  all  the  figures  are  per  cents  and 
that  100  per  cent  is  unity  or  1.00,  while  96.32  per  cent  is  .9632,  etc.  That  is,  before  mul- 
tiplying percentages,  we  must  shift  the  decimal  point  two  places  to  the  left;  and,  of 
course,  after  obtaining  the  result  (e.g.  1.9342  for  1918),  we  must  shift  the  decimal  point 
back  again  (i.e.  for  1918,  193.42). 

Graphically,  in  charting  price  movements,  each  index 
number  is  represented  by  a  point  high  or  low  in  the  dia- 
gram acc6rding  as  the  index  number  is  large  or  small. 
Thejyhole  series  of  points  for  different  dates,  whether 
each/jpoint  is  obtained  by  the  fixed  base  method  or  by  the 
chain  of  bases  method,  may  be  joined  together,  forming  a 
curve.  The  picture  of  the  simple  arithmetic  index  num- 
ber relative  to  1913  as  a  fixed  base  is  given  in  Chart  5P 
(curve  labeled  "1").  The  " chain"  figures,  relative,  in- 
directly, to  1913,  are  indicated  by  small  balls  which  come 
sometimes  above  and  sometimes  below  the  original  curve 
calculated  by  the  " fixed  base"  method.  There  are  no 
balls  for  the  year  1914  as  the  two  numbers  for  that  year 
are,  of  course,  identical. 

This  graphic  system  of  distinguishing  the  results  of  the 
" fixed  base"  and  " chain  base"  methods  of  working  an 


24  THE  MAKING  OF  INDEX  NUMBERS 

Simple  Arlfhmet/c  Index  Number  of  Prices 

of    36  Commodities 
Compared  wifh  "No.  353" 


5% 


;j  74  75  76  77  76 

CHART  5P.  Comparison  of  two  index  numbers  of  prices  of  the  36  com- 
modities, by  Formula  No.  1  (simple  arithmetic)  and  Formula  No.  353  (the 
"  ideal  V  as  later  explained).  Each  of  the  points  joined  by.  lines  is  relative 
directly  to  the  fixed  base  (1913),  and  each  small  ball  is  relative  indirectly 
to  1913  via  intermediate  years  (i.e.  relative  directly  to  the  preceding  small 
ball  as  base,  which  in  turn  is  likewise  relative  to  its  preceding  ball,  and 
so  on  back  to  1913). 

index  formula  will  be  used  throughout  the  following  in- 
vestigation so  that  the  " fixed  base"  and  " chain  base" 
results  may  be  compared  on  various  charts  for  all  the 
six  types  of  formulae  —  arithmetic,  harmonic,  geometric, 
median,  mode,  aggregative.  In  the  case  of  the  simple 
arithmetic  index  number  there  is  evidently  an  appreciable 
discrepancy  between  the  fixed  base  and  the  chain  figures. 

§  7.  Aids  to  Interpreting  the  Charts 

To  interpret  such  curves  as  the  foregoing  and  those 
which  follow,  it  will  help  the  reader  to  note  carefully  the 
heights  representing  an  increase  of  one  per  cent,  five  per 
cent,  etc.  In  Chart  6P,  for  instance,  the  length  of  the 
dark  vertical  line  marked  "5  %"  (as  noJLed  under  Chart 
3P)  affords  a  visual  measuring  rod  by  which  it  is  possible 
to  get  a  clear  idea  of  the  percentage  by  which  any  given 


SIX  TYPES  OF  INDEX  NUMBERS 


25 


point  in  any  diagram  in  this  book  is  higher  than  any  other 
point,  all  the  diagrams  being  drawn  on  the  same  scale. 
In  Chart  6P  the  application  of  this  measuring  rod  to  the 
slopes  of  the  lines  is  indicated  in  another  way.  Each  of 
the  short  lines  lying  above  the  curve  ascends  in  a  year 


Index  No.  353    of  Prices 

contrasted 

(/)  with  dotted  lines  above    each  diverging 
sx  in  a  year, 

(2)  with  dotted  lines  below,  each  diverging 
i%  in  o  year 


355 


'/J  14  V5  76  17  78 

CHART  6P.   An  aid  to  the  eye  for  judging  contrasts  in  subsequent  charts. 

yfive  per  cent  more  than  the  corresponding  line  in  the 
curve,  while  each  of  the  short  lines  below  the  curve  as- 

"-cends  one  per  cent  more  than  the  corresponding  line  in 
•the  curve. 

Chart  7  will  also  help  in  future  interpretations  of 
curves.*  By  the  method  of  plotting  here  used  (called  the  • 
ratio_chart  method 1),  the  line  representing  a  uniform  per- 
centage of  change,  say  ten  per  cent  per  year,  will  simply  | 
go  on  being  straight.     Thus,  if  an  index  number  increases 
in  the  first  year  ten  per  cent,  that  is,  from  100  to  110,  and 

1  For  a  full  discussion  of  the  advantages  of  this  method  see  Irving  Fisher, 
"The  Ratio  Chart,"  Quarterly  Publications  of  the  American  Statistical  As- 
sociation, vol.  xv  (1917),  p.  577.  The  method  is  also  called  "logarithmic." 


26 


THE  MAKING  OF  INDEX  NUMBERS 


Uniform  Slopes* Uniform  Ratios 


70 

e&s 

CHART  7.  Showing  the  fundamental  feature  of  the  "ratio  chart"  method 
used  throughout  this  book,  namely,  the  uniform  significance  of  direction. 
The  upper  line  representing  a  continuous  series  of  equal  percentage  increases 
(100  to  110  is  10  per  cent;  110  to' 121  is  10  per  cent;  200  to  220  is  10  per 
cent)  is  straight^  T^ejthree  lower  lines  are  parallel  to  each  other,  one  rep- 
resentingjthe  actual  prices  $1.20  and  $1.80,  one  representing  the  price  rela- 
tives starting  with  100  per  cent,  and  the  other  the  price  relatives  ending 
with  100-per  qeixb. 


SIX  TYPES  OF  INDEX  NUMBERS  27 

likewise  ten  per  cent  in  the  second  year,  that  is,  from  110 
to  121,  and  so  on,  increasing  each  year  ten  per  cent  (in 
the  last,  from  200  to  220),  it  will  simply  continue  its 
straight  course,  the  rises  of  10,  11,  .  .  .  ,  and  20  all  being 
equal  percentage  rises  (though  not  equal  differences). 

It  further  follows  that  any  two  lines  representing  equal 
percentage  rates  of  change  will  be  parallel.  Thus,  if  a 
commodity  changes  in  price  from  $1.20  per  bushel  to 
$1.80  per  bushel,  or  50  per  cent,  the  line  representing 
this  change  in  actual  prices  will  be  parallel  to  a  line 
representing  merely  their  relative  changes  from  100  per 
cent  to  150  per  cent  and  parallel  also  to  a  line  repre- 
senting the  reverse  relative  changes  from  100  per  cent 
backward  to  66 f  per  cent. 

The  central  curve  (Chart  6P)  might  have  been  any  curve. 
As  a  matter  of  fact  it  is  the  curve  obtained  by  using  the 
formula  called  353  in  this  book  (the  calculations  being 
relative  to  1913  as  a  fixed  base).  Since  Formula  353  is 
the  one  which  we  shall  find  to  be  the  best,  —  the  " ideal" 
one  —  the  reader  may  care,  for  the  sake  of  future  com- 
parisons, to  establish  at  the  outset  a  mental  picture  of 
this  curve. 


§  8.  The  Algebraic  Formula  for  the  Simple  Arithmetic 
Index  Number 

Algebraically,  the  formula  for  the  simple  arithmetic 
average  was  previously  given  for  two  and  for  three  com- 
modities. For  36  commodities  the  formula  for  1914  as 
year  "1"  (relatively  to  1913  as  the  base  year,  or  year 
"0")  is  evidently 


i     /         //         /// 
Po       p  o       P    o       P     o 

36 


28  THE  MAKING  OF  INDEX  NUMBERS 

In  order  to  avoid  writing  so  many  terms  the  best  usage 
is  to  call  the  numerator  sf^lH  where  the  symbol   "S" 


k  is  the  Greek  letter  Sigma  or  "S,"  the  initial  letter  of 
"  Sum."  It  does  not  denote  a  quantity,  but  is  an  abbrevi- 
ation for  the  words  "the  sum  of  terms  like  the  following 

^sample"  so  that  the  above  expression,  written  with  this 

^convenient  abbreviation  for  summation,  is 


or,  more  generally, 

'$)         f<-      '.     -*- 

^A'O7 


71 

/ 


n  stands  for  the  number  of  commodities,  whether 
rthis  be  36  or  any  other  number. 

Just  as  p  stands  for  price,  so  we  may  let  q  stand  for 
quantity  (bushels,  etc.).  The  simple  arithmetic  index 
number  for  the  quantities  of  the  36  commodities  would, 
therefore,  be 

* 


n 

Similarly  the  formula  for  1 91 5  (year  "2")  relatively 
to  1913  (year  "0")  is,  for  prices, 


V) 

wJ 
and,  for  quantities, 


21 
n 


2(2! 


n 


SIX  TYPES  OF  INDEX  NUMBERS  29 

Again,  by  replacing  "2"  with  "3"  we  have  the  formulae 
for  1916.  Likewise  those  for  1917  and  1918  are  obtained 
by  similarly  substituting  "4"  and  "5." 

Turning  from  these  fixed  base  formulae  to  the  chain 
system,  we  first  note  that  the  formula  for  the  simple 
arithmetic  index  number  of  prices  for  1915  relatively  to 
1914  —  that  is,  the  second  link  in  the  chain  system  —  is 
evidently 


<£) 


n 
Since  the  formula  for  1914  relatively  to  1913  is 


<£) 


formula  for  1915  relatively  to  1913  via  1914  is  the 
product  of  the  two  preceding  expressions;  likewise,  the 
chain  formula  for  1916  is  the  product  of  three  such  ex- 
pressions,  and  so  on  for  any  number  of  links. 

§  9.  The  Simple  Arithmetic  —  Usage  and  Utility 

The  simple  arithmetic  average  is  perhaps  still  the  fa- 
vorite one  in  use.  It  was  used  as  early  as  1766  by  Carli.1 
It  is  used  by  the  London  Economist,  the  London  Statist 
(continuing  Sauerbeck's  index  number)  and  many  other 
makers  of  index  numbers. 

In  the  present  exposition,  the  simple  arithmetic  average 
is  put  first  merely  because  it  naturally  comes  first  to  the 
reader's  mind,  being  the  most  common  form  of  average. 
In  fields  other  than  index  numbers  it  is  often  the  best 
form  of  average  to  use.  But  we  shall  see  that  the  simple  i 
arithmetic  average  produces  one  of  the  very  worst  of 

1  See  C.  M.  Walsh,  The  Measurement  of  General  Exchange  Value,  p.  534. 


30  THE  MAKING  OF  INDEX  NUMBERS 

index  numbers.    And  if  this  book  has  no  other  effect 
s   than  to  lead  to  the  total  abandonment  of  the  simple  arith- 
metic type  of  index  number,  it  will  have  served  a  useful 
purpose. 

The  simple  arithmetic  index  number  just  described  is 
listed  in  the  Appendix  as  Formula  1  and  will  often  be 
^  referred  to  by  that  identification  number. 

§10.  The  Simple  Harmonic 

The  next  simple  index  number  to  be  explained  is  the 
harmonic,  the  identification  number  of  which  in  this 
book  is  11.  (The  numbers  between  "1"  and  "11"  will 
Jbe  assigned  to  other  formulae  to  be  introduced  later.) 

..The  process  of  calculating  the  simple  harmonic  average 
^is  somewhat  like  that  of  calculating  the  simple  arithmetic, 
^differing  merely  in  that  reciprocals  are  employed.     The 
term  "reciprocal"  is  here  used  in  the  mathematical  sense, 
the  reciprocal  of  any  number  being  the  quotient  ob- 
tained by  dividing  unity  by  that  number.     If  the  number 
is  expressed  in  fractional  form,  its  reciprocal  is  found 
by  turning  the  fraction  upside  down.     Thus  the  reciprocal 
ofS  (i.e.  f )  is  i ;  the  reciprocal  of  3  (i.e.  f )  is  J ;  of  i  is  £ ,  etc. 

There  are  three  steps  in  calculating  the  simple  harmonic 
average  of  any  given  set  of  ratios : 
_-  (1)   turn  the  ratios  upside  down ; 
— (2)   take  the  simple  arithmetic  average  of  these  in- 
verted figures ; 

—(3)   turn  the  average  thus  obtained  right  side  up  again. 
Thus  to  take  the  simple  harmonic  average  of  f  and  $• : 

(1)   their  upside  down  ratios,   or  "reciprocals,"  are  f 
andf; 
s     (2)   the  simple  arithmetic  average  of  the  last  two  is  V  5 

(3)   the  reciprocal  of  the  last  is  -fr  or  H,  which  is  the 
desired  simple  harmonic  average. 


SIX  TYPES  OF  INDEX  NUMBERS 


31 


This  harmonic  average  of  f  and  f  (which  is  M )  is  less  than 
tjie  simple  arithmetic  average  of  f  and  f,  which  is  H. 

Let  us  apply  this  process  to  index  numbers.  It  is  the 
second  process  —  averaging,  the  percentaging  being  al- 
ready done.  Taking,  then,  the  36  price  relatives  indicated 
in  Table  1  above,  viz-.,  104.77  per  cent,  i.e.  1.0477,  .9906, 
.  .  .  ,  tp  1.1152  (the 36th),  then  inverting  them  into  .9545, 
1.0095,.  .  .  ,  to  .8967;  then  taking  the  simple  arithmetic 
average  of  these,  which  is  1.0506 ;  then  inverting  the  latter, 
we  ge't  finally  .9519  or  95.19  per  cent,  which  is  the  simple 
harmonic  index  number.  This  is  less  than  the  simple 
arithmetic  index  number  for  the  same  year  (96.32  per 
cent)  already  found. 

The%  complete  series  of  simple  harmonic  index  numbers 
of  prices,  both  by  the  fixed  base  and  the  chain  system, 
are  given  below  and  also,  for  comparison,  the  simple 
arithmetic  by  the  same  two  methods. 


FORMULA  No. 

TYPE 

BASE 

1913 

1914 

1913 

1916 

1917 

1918 

1 

Simple  arithmetic 

Fixed  - 
Chain 

•100. 
100. 

96.32 
-96.32 

98.03 
,97.94 

123.68 
125.33 

175.79 
175.65 

186.70 
193.42 

11 

Simple  harmonic 

Fixed- 
Chain_ 

100. 
100. 

,95.19 
/95.19 

95.58 
/8S.64 

^119.12 
^17.71  ( 

r57.88 
,158.47 

<171.79 
,•167.76 

It  will  be  noted  that  there  are  great  differences  here 
among  the  results  of  the  four  methods,  especially  in  1917 
and  1918 ;  and  that  the  harmonic  is  always  less  than  the 
arithmetic.  The  reason  for  this  need  not  be  considered  here, 
the  harmonic  index  number  (fixed  base) 
s  given  in  Charts  SP  and  SQ  (Curve  11)  with  all  the  five 
other  simple  index  numbers,  —  arithmetic,  geometric, 
median,  mode,  aggregative.  As  in  the  case  of  the  chain 
arithmetics  and  fixed  base  arithmetics,  the  chain  har- 
monics do  not  agree  with  the  fixed  base  harmonics. 


32  THE  MAKING  OF  INDEX  NUMBERS 

Simple  Index  Numbers  of  Prices 


75 


75 


16 


77 


CHART  8P.  In  the  upper  group,  the  simple,  geometric  (21)  necessarily 
lies  between  the  simple  arithmetic  (1)  above  it  and  the  simple  harmonic 
(11)  below  it.  Of  the  lower  group,  the  simple  median  (31)  most  resembles 
the  upper  group,  while  the  simple  mode  (41)  and  simple  aggregative  (51) 
are  each*  sui  genesis.  The  two  groups  are  separated  to  save  confusion, 
really  forming  two  distinct  diagrams. 

f      Algebraically,  the  simple  harmonic  average  of  the  two 
\  price  ratios,  2l  and  2-1,  is  the  reciprocal  of  the  arithmetic 

Plo  P  0 

average  of  their  reciprocals  ;  that  is, 

2 


For  three  terms  the  formula  is 

3 


/      ^// 


Pi 


For  n  terms  the  formula  is 


n 


Pi 


SIX  TYPES  OF  INDEX  NUMBERS  33 

Simple  Index  Numbers  of  Quantities 


73  74  75  75  77 

CHART  8Q.  Analogous  to  Chart  8P. 


73 


The  harmonic  index  number  has  found  few  champions. 
One  of  these  is  F.  Coggeshall.1  We  shall  find,  however, 
that  the  simple  harmonic  is  a  sort  of  " antithesis"  of  the 
simple  arithmetic;  and  when  we  arrive  at  their  faults 
we  shall  find  the  two  equally  at  fault  but  in  opposite 
directions. 

§  11.  The  Simple  Geometric 

We  now  come  to  the  simple  geometric  index  number. 
The  reader  whose  conception  of  an  average  has  been 
limited  to  the  arithmetic  is  referred  to  Appendix  I  (Note 
A  to  Chapter  II,  §  15)  for  a  general  definition  of  average 
which  will  include  the  harmonic  and  the  others  used 
below.  Suffice  it  here  to  define  the  geometric  average. 

Given  the  price  relatives,  in  order  to  get  the  average 
of  them  (that  is,  the  index  number)  by  the  simple  geo- 

1  F.  Coggeshall,  "The  Arithmetic,  Geometric,  and  Harmonic  Means," 
Quarterly  Journal  of  Economics,  vol.  1  (1886-87),  pp.  83-86. 


34  THE  MAKING  OF  INDEX  NUMBERS 

metric  formula  (21  in  our  series),  instead  of  adding  to- 
gether the  price  relatives  of  the  listed  commodities  and 
then  dividing  their  sum  by  the  number  of  terms  (n)  we 
|  multiply  the  price  relatives  together  and  then  extract  the 
\  nth  root. 

Thus  to  get  the  simple  geometric  average  of  2  and  8, 
we  take  their  product  (16)  and  extract  its  square  root, 
obtaining  V2  X  8  =  4.  To  get  the  simple  geometric 
average  of  the  three  numbers  4,  6,  and  9,  we  take  their 
product  (216)  and  extract  the  cube  root,  obtaining  6  as 
the  simple  geometric  average.  To  get  the  simple  geo- 
metric average  of  the  four  numbers  3,  4,  6,  and  18,  we 
take  their  product  (1296)  and  extract  the  fourth  root, 
obtaining  6. 

Numerically,  to  apply  the  geometric  process  to  index 
numbers,  we  multiply  all  the  36  price  relatives,  104.77 
per  cent,  99.06  per  cent,  .  .  .  ,  111.52  per  cent,  together 
and  extract  the  36th  root,  a  process  made  easy  by  means 
of  logarithmic  tables.1  The  result  is  95.77  per  cent  for 
1914  relatively  to  1913,  whereas  the  simple  arithmetic 
method  gave  96.32^  and  the  simple  harmonic,  95.19  per 
cent.  The  geometric  will  be  found  to  lie  between  the 
arithmetic  (which  is  always  above  it)  and  the  harmonic 
(which  is  always  below  it). 

Graphically,  the  geometric  index  number  is  given  in 
Chart  8  (Curve  21)  with  all  the  five  other  simple  index 
numbers,  —  arithmetic,  harmonic,  median,  mode,  aggre- 
gative. 

Algebraically,  the  simple  geometric  average  of  two  price 
ratios  is  given  by  the  formula 


Po      Pa 

1  For  model  examples  to  aid  in  the  practical  calculation  of  this  as  well  as 
of  eight  other  sorts  of  index  numbers,  see  Appendix  VI,  §  2. 


SIX  TYPES  OF  INDEX  NUMBERS,  35 

For  three,  the  formula  is 


Po     P  o     P   o 
For  any  number,  ny  it  is 


X  ...     (n  terms). 
Po     po     p   o 

In  the  case  of  the  simple  geometric  average  the  "chain" 
figures  are  always  identical  with  those  calculated  rela-  w^ 
tively  to  a  fixed  base.1 

Jevons,2  in  1863,  used  and  advocated  the  simple  geo- 
metric. It  still  finds  some  favor  among  statisticians 
and,  as  we  shall  see,  really  deserves  a  high  place  among 
the  simple  averages,  when  simple  averages  are  called  for. 
But  whether  the  fact  that  the  chain  figures  agree  with 
the  fixed  base  figures  is  a  virtue  will  be  discussed  in  Chap- 
ter XIII. 

§  12.  The  Simple  Median 

The  simple  median  (Formula  31)  is  calculated,  not  by 
the  processes  of  adding  and  dividing,  or  of  multiplying 
and  extracting  a  root,  but  merely  by  selecting  the  middle- 
most term.  Thus,  the  median  of  3,  4,  and  5  is  evidently  4, 
the  middle  term.  The  median  of  1,  3,  3,  4;  4,  4,  5,  6,  6,  6, 
6,  7,  7  is  5,  since  5  stands  in  the  middle  of  the  list,  there 
being  six  items  smaller  and  six  items  larger.  The  median 
height  of  a  line  of  51  soldiers  standing  in  the  order  of  their 
heights  is  the  height  of  the  middlemost  soldier,  i.e.  the 
26th  from  either  end. 

When  the  number  of  terms  is  even,  there  are  two  middle- 
most terms  instead  of  one.  If  these  two  are  alike  either 
of  them  may,  of  course,  be  called  the  median.  If  the  two 

1  For  proof,  see  Appendix  I  (Note  to  Chapter  II,  §  11). 

2  See  Walsh,  The  Measurement  of  General  Exchange  Value,  p.  557. 


36 


THE  MAKING  OF  INDEX  NUMBERS 


middle  terms  differ,  then  the  median  lies  between  them 
and  cannot  be  definitely  determined  without  recourse  to 
some  other  process  of  averaging  such,  for  example,  as  tak- 
ing the  simple  arithmetic  or  simple  geometric  average  of 
the  two  middle  terms. 

By  the  fixed  base  method  (recurring  to  our  36  com- 
modities), the  median  of  the  price  relatives  of  1914  (rela- 
tively to  1913)  is  99.45,  and  that  of  1915  (also  relatively 
to  1913)  is  98.57.1  By  the  chain  method,  the  median 
for  1915  (relatively  to  1913  via  1914)  becomes  99.33.  The 
two  methods  under  the  median  are  compared  below : 


FORMULA 
No. 

BASE 

1913 

1914 

1915 

1916 

1917 

1918 

31 

Fixed 

100. 

99.45 

98.57 

118.81 

163.81 

190.92 

31 

Chain 

100. 

99.45 

99.33 

117.50 

155.86 

180.07 

Graphically,  Chart  8  (Curve  31)  shows  the  median  as 
well  as  the  five  other  simple  index  numbers. 

Professor  Edgeworth  (1896)  recommended  the  simple 
median.  Since  this  advocacy  several  statisticians  have 
used  it,  including  A.  L.  Bowley  and  Wesley  C.  Mitchell. 

§  13.  The  Simple  Mode 

The  simple  mode  (Formula  41)  is  found  by  arranging 
the  items  in  order  of  size,  just  as  in  the  case  of  the  median, 
and  then  selecting,  not  the  middlemost  term,  but  the  com- 
monest term;  hence  the  word  "mode"  indicating  "most 
in  vogue."  Thus  the  mode  of  1,  2,  3,  3,  4,  4,  4,  5,  5,  5,  5, 
6,  6,  7  is  5 ;  for  5  occurs  four  times  while  no  other  number 
occurs  oftener  than  three  times. 

But,  even  more  than  the  median,  the  mode  is  ambig- 

1  For  model  examples  to  aid  in  the  practical  calculation  of  this  as  well  as 
eight  other  sorts  of  index  numbers,  see  Appendix  VI,  §  2. 


SIX  TYPES  OF  INDEX  NUMBERS  37 

uous  and  vague  and  needs  to  be  helped  out  by  other  pro- 
cesses than  that  given  in  its  own  definition.  Ordinarily, 
few  of  the  items  are  just  alike  so  that,  to  make  the  mode 
a  workable  average,  we  do  not  really  count  the  repetition 
of  precisely  equal  terms  but  the  repetition  of  terms  falling 
within  hailing  distance  of  each  other,  or,  more  precisely, 
within  certain  arbitrarily  chosen  limits. 

Thus,  for  the  line  of  soldiers,  we  should  probably  not 
find  any  two  of  exactly  the  same  height;  but  we  could 
easily  classify  them  in  groups  differing  by  inches.  At 
one  end  of  the  line  are  the  short  men  between,  say,  5  feet 
6  niches  and  5  feet  7  inches  of  which  there  may  be  only, 
say,  two  soldiers.  Let  us,  for  convenience  of  thought, 
imagine  these  to  be  set  apart  in  a  group  by  themselves, 
separated  a  little  from  the  next  taller  group  contain- 
ing, say,  five  soldiers  between  the  heights  5  feet  7  inches 
and  5  feet  8  inches,  and  these  in  turn  separated  from  those 
within  the  next  inch  (5  feet  8  inches  to  5  feet  9  inches) 
numbering,  say,  20  soldiers.  Within  the  next  inch  of 
height  (5  feet  9  inches  to  5  feet  10  inches)  are,  say,  30 
soldiers;  the  next  (5  feet  10  inches  to  5  feet  11  inches), 
25 ;  the  next  (5  feet  11  inches  to  6  feet),  10. 

Evidently  here  the  commonest  height  is  that  of  the 
group  (of  30  soldiers)  between  5  feet  9  inches  and  5  feet 
10  inches,  which  is,  therefore,  the  mode.  To  put  any 
finer  point  on  it,  i.e.  to  find  the  mode  any  more  closely 
than  within  an  inch  would  require  either  subdividing  by 
half-inch  intervals,  or  else  mathematically  or  graphically 
adjusting  the  figures  so  as  to  make  a  " smooth"  curve  of 
frequency  and  then  taking  the  maximum  on  this  ideal 
curve  to  represent  the  mode,  or  resorting  to  some  other 
extraneous  aid. 

The  best  example  of  the  mode  as  applied  to  index 
numbers  is  that  afforded  by  the  "  Summary  of  the  History 


38  THE  MAKING  OF  INDEX  NUMBERS 

of  Prices  during  the  War"  (Bulletin  No.  1,  of  the  War 
Industries  Board)  by  Wesley  C.  Mitchell.  Thus,  to 
take  the  mode  of  1918:  Out  of  1437  commodities,  the 
prices  of  which  were  reckoned  relatively  to  the  pre-war 
year  as  a  base,  there  were  two  commodities  the  prices  of 
which  were  between  30  and  49  per  cent  of  the  pre-war 
prices;  four  between  50  and  69  per  cent;  17  between 
70  and  89  per  cent,  and  for  the  succeeding  similar  in- 
tervals of  20  each,  the  following  successive  numbers  of 
items,  namely :  61  items,  64  items,  130,  212,  219,  164,  135, 
104,  76,  54,  42,  30,  31,  16,  13,  7,  7,  8,  4,  4,  4,  5,  3,  4,  1, 
0,  1,  0,  0,  0,  0,  1,  2,  1,  1,  0,  1,  1,  0,  0,  1,  etc.  The  price  of 
the  last  named  solitary  commodity  was  between  890  and 
909  per  cent  of  its  pre-war  price.  The  mode  here  lies  in 
the  compartment  having  the  largest  number  (219).  This 
compartment  is  that  between  170  and  189.  The  mode 
lies,  therefore,  somewhere  between  170  per  cent  and  189 
per  cent.  The  exact  location  of  the  mode  is  always 
more  or  less  mythical.  In  this  case  173  is  obtained  by  a 
graphical  method  as  the  value  of  the  mode.  To  put  any 
finer  point  on  it  would  be  almost  meaningless. 

Thus  the  median  and  the  mode  are  both  somewhat 
indeterminate,  the  mode  especially  so  in  case  of  wide  and 
irregular  dispersion  of  price  relatives  unless  the  number 
of  items  runs  into  the  hundreds  or  thousands. 

Numerically,  in  the  case  of  the  36  commodities  the 
mode  for  1914  (relatively  to  1913)  graphically  obtained, 
was  98. *  Another  method  (calculating  it  indirectly  from 
the  simple  arithmetic  and  the  simple  median)  makes  it 
106  and  another  similar  method,  applied  in  the  reverse 
direction,  makes  it  109.  But  when  dealing  with  so  few 
commodities  as  36  the  mode  is  so  indeterminate  that  it  is 
not  worth  while  to  employ  it.  For  completeness,  how- 
1  See  Appendix  I  (Note  to  Chapter  II,  §  13). 


SIX  TYPES  OF  INDEX  NUMBERS  39 

ever,  the  mode  (as  calculated  by  a  graphic  method)  has 
/been  entered  in  the  tables,  although  omitted  from  most 
/of  the  charts. 

By  the  chain  method,  the  mode  for  1915,  relatively 
^  to  1913,  via  1914,  is  roughly  95 ;  by  the  fixed  base  method 
/it  is  98.     The  figures  for  all  the  years  are  given  in  Appen- 
.    dixVII. 

Graphically,  the  simple  mode  (Curve  41)  with  the  five 
'  other  simples  is  given  in  Chart  8. 

+/    The  mode  has  never  been  either  used  or  proposed  for 

use  in  index  numbers.     But  Wesley  C.  Mitchell,  in  Bulle- 

^  tin  1 73 j  and  its  revised  edition  284,  °f  ^ne  United  States 

^Bureau  of  Labor  Statistics,  and  (as  has  just  been  noted) 

„    in  his  War  Industries  Board  "  History  of  Prices,"  has 

^presented  some  figures  to  illustrate  the  mode,  as  have 

some  other  writers.     Mr.   C.  M.  Walsh  has  suggested 

that  the  position  of  the  mode  in  relation  to  the  arithmetic 

and  other  averages  may  help  us  select  the  best  average 

to  use.     But  even  this  supposed  utility  of  the  mode  will 

be  found  illusory. 

§  14.   The  Simple  Aggregative 

Last  of  the  simple  index  numbers  is  the  simple  aggre- 
gative^ (Formula  51).  This  is  the  percentage  obtained 
>y  taking  the  aggregate,  or  sum,  of  all  the  actuaLprices 
for  a  given  year  and  dividing  this  by  the  sum  of  the  prices 
for  the  base  year.  Thus,  while  the  arithmetic  starts  off 
by  adding  relative  prices,  the  aggregative  starts  off  by 
adding  agfajaTpriqes. 

Numerically,  the  sum  of  all  the  prices  in  1913  (i.e.  12.36 
+  62.63  +  .  .  .  +  37.58)  is  23889.48  and  the  sum  of 
all  the  prices  in  1914  (i.e.  12.95  +  62.04  +  .  .  .  +  41.91) 
is  22905.24,  so  that  the  simple  aggregative  index  number  is 

22905  94 

>  or  95. 88  per  cent.  Under  the  simple  aggregative 


40  THE  MAKING  OF  INDEX  NUMBERS 

formula,  the  chain  figures  and  the  fixed  base  figures  are 
identical,  as  is  evident.1 

Graphically,  Chart  8  gives  the  simple  aggregative  (Curve 
51)  with  the  five  other  simples. 

Algebraically  ,  the  formula  for  the  aggregative  index 
number  is 

Pi  +  p'i  +  p"i  +  .  .  . 


PQ  +  P'Q  +  P"O  + 
or,  more  briefly, 


We  have  seen  that  to  get  the  simple  aggregative  index 
number  we  do  not  first  calculate  price  relatives  at  all; 
we  use  the  orignaljnices.  In  fact,  unlike  the  other  types 
of  index  numbers,  the  simple  aggregative,  being  the  ratio 
of  sums  or  aggregates  of  prices,  cannot  be  calculated  from 
the  price  relatives  alone.  It  requires  the  actual  prices 
themselves.  It  would  not  be  enough  to  know  that  the  price 
of,  say,  sugar  was  twice  what  it  was  at  the  base  date  in 
order  to  be  able  to  calculate  the  aggregative  index  num- 
ber. We  would  need  to  go  back  to  the  actual  prices  of 
sugar  at  the  two  dates  —  whether,  for  instance,  6  cents 
and  12  cents  respectively,  or  some  other  pair  of  figures  in 
the  same  proportion,  such  as  8  cents  and  16  cents. 

The  simple  aggregative  index  number  is  usually  re- 
garded as  almost  worthless  ;  and  so  it  is,  unless  the  units 
i  of  measurement  are  discreetly  chosen. 

The  aggregative  form  of  an  index  number  was  used  as 
early  as  1738  by  Dutot.2  The  only  conspicuous  instance 
of  its  actual  use  is  in  Bradstreet's  index  number  where  the 
prices  are  first  reduced  to  prices  per  pound  for  every  item. 

1  See  Appendix  I  (Note  to  Chapter  II,  §  14). 

2  See  C.  M.  Walsh,  The  Measurement  of  General  Exchange  Value,  p.  534. 


SIX  TYPES  OF  INDEX  NUMBERS  41 

§  15.  The  Six  Simple  Index  Numbers  Compared 

In  our  tables  the  simple  index  mfmbers,  namely,  'the 
simple  arithmetic,1 'simple  harmonic,  *  simple  geometric, 
V  simple   median,  Isimple   mode,   and  ^  simple  aggregative 
index  numbers,  have,  as  already  stated,  as  their  identi- 
fication numbers  1,  11,  21,  31,  41,  51,  respectively. 

These  six  types  represent  six  different  processes  of  cal- 
culation, namely:   for  (1),  adding  the  price  relatives  to- 
gether and  dividing  by  their  number;   for  (11),  adding 
/their  reciprocals  together  and  dividing  into  their  number ; 
for  (21),  multiplying  the  price  relatives  together  and  ex- 
v^actirig  the  root  indicated  by  then1  number;    for  (31), 
arranging  the  price  relatives  in  order  of  size  and  selecting 
/he  middlemost;  for  (41),  so  arranging  but  selecting  the 
/fommonest;   for  (51),  adding  together  the  actual  prices 
of  each  year  and  taking  the  ratio  of  these  sums.1 

Graphically,  Chart  8  gives  all  the  six  simple  index  num- 
bers, both  of  prices  and  quantities,  corresponding  to 
Formulae  1,  11,  21,  31,  41,  51.  Curves  1,  11,  21  are  drawn 
from  a  common  origin,  separately  from  the  others,  be- 
cause they  are  interrelated,  No.  21  always  lying  between 
1  and  II.2 

"As  to  the  other  three,  the  median  lies,  with  one  trifling 
exception,  above  the  mode.  This  is  not  a  law  but  is  apt 
to  be  the  case  when,  as  in  the  present  example,  the  items 
averaged  are  more  widely  dispersed  upward  than  down- 
ward, for  downward  dispersion  is  limited  by  the  existence 
of  a  zero  below  which  prices  and  quantities  cannot  sink. 

The  simple  aggregative  is  a  law  unto  itself,  by  reason 
of  its  peculiar  and  haphazard  weighting.  I  have  called 

1  For  a  general  definition  of  average  covering  these  six  and  others  see 
Appendix  I  (Note  A  to  Chapter  II,  §  15). 

2  See  Appendix  I  (Note  B  to  Chapter  II,  §  15). 


42  THE  MAKING  OF  INDEX  NUMBERS 

it  "simple,"  but  it  is  not  simple  in  quite  the  same  sense 
as  are  the  other  five.  As  Walsh  says,  it  is  "  haphazard/ ' 
being  dependent  on  the  accident  of  what  measures  or 
units  are  used  in  pricing  the  commodities  in  the  list.  If 
silver,  instead  of  being  quoted  per  ounce  (as  it  was  in 
computing  this  average  because  the  ounce  is  the  usual 
unit  used  in  published  silver  quotations),  had  been  quoted 
in  tons,  and  if  coal  had  been  quoted  in  ounces,  instead 
of  tons,  the  result  would  be  entirely  different.  Silver 
would  dominate,  and  the  average  curve  would  nearly 
coincide  with  the  silver  curve  (in  Chart  3),  while  coal 
would  have  a  negligible  influence. 

It  must  be  admitted  that  this  first  view  of  the  six  differ- 
ent types  of  index  numbers  is  not  reassuring.  If  one  of 
these  indexes  were  as  good  as  another,  then  certainly 
they  would  all  be  almost  good  for  nothing ;  for  they  dis- 
agree with  each  other  very  widely  indeed,  both  when  com- 
puted for  a  fixed  base  and  when  computed  through  a 
chain  of  bases.  The  lowest  index  number  for  1917  (that 
by  Formula  41)  is  135,  and  the  highest  (that  by  Formula 
1)  is  175.79,  which  latter  is  30  per  cent  above  the  former. 
While  this  range  is  much  less  than  the  divergence  of  the 
individual  price  relatives  themselves,  it  is  altogether  too 
great  to  make  possible  any  statistics  worthy  of  the  name. 
All  that  could  be  claimed  is  that,  where  there  is  not  so 
wild  a  dance  of  prices  as  injbhe  war  years,  the  six  types 
of  averages  will  themselves  be  less  discordant.  But, 
fortunately  for  the  science  olindejiL  numbers,  the  six  types 
do  not,  as  we  shall  see,Jiaye  equal  daims. 


CHAPTER  III 
FOUR  METHODS  OF  WEIGHTING 

§  1.  Weighting  in  General 

IT  has  already  been  observed  that  the  purpose  of  any 
index  number  is  to  strike  a  "  fair  average  "  of  the  price  move- 
ments—  or  movements  of  other  groups  of  magnitudes.1 
At  first  a  simple  average]  seemed  fair,  just  because  it 
treated  all  terms  alike.  And,  in  the  absence  of  any  knowl- 
edge of  the  relative  importance  of  the  various  commodi- 
ties included  in  the  average,  the  simple  average  is  fair. 
But  it  was  early  recognized  that  there  are  enormous  dif- 
ferences in  importance.  Everyone  knows  that  pork  is 
more  important  than  coffee  and  wheat  than  quinine. 
Thus  the  quest  for  fairness  led  to  the  introduction  of 
weighting.  At  first  the  weighting  was  rough  and  ready, 
being  based  on  guesswork.  Arthur  Young  called  barley 
twice  as  important  as  wool,  coal,  or  iron,  while  he 
called  " provisions"  four  times  as  important,  and  wheat 
and  day  labor  each  five  times  as  important. 

But  what  is  the  just  basis  for  assigning  weights?  Arbi- 
trary weighting  may  be  an  improvement  over  a  simple 
index  number ;  but,  if  abused,  it  may  aggravate  the  un- 
fairness. If  we  werfc  deliberately  to  seek  the  most 
/  unfair  weighting,  we  could  give  any  one  commodity  so 
preponderate  a  weight  as  to  make  the  resulting  index 
number  practically  follow  the  course  of  that  particular 
commodity. 

1  "Purchasing  power"  included,  although  not  explicitly  treated  in 
this  book.  See  Appendix  I  (Note  to  Chapter  III,  §  1). 

43 


44  THE  MAKING  OF  INDEX  NUMBERS 

To  cite  an  extreme  example,  take  the  1366  commodities 
in  the  carefully  weighted  index  number  of  the  War  In- 
dustries Board.  According  to  this  excellent  index  num- 
ber, prices  rose  between  the  pre-war  year  (i.e.  the  year 
from  July  1,  1913,  to  July  1/1914),  and  the  calendar 
year  1917  in  the  ratio  of  100  to-175.  This  figure  is  a  very 
representative  of  the  1366  figures  from  which  it  was 
calculated,  although  these  range  all  the  way  from  a  price 
relative  of  35  for  oil  of  lemon  to  a  price  relative  of  3910 
for  potassium  permanganate.  But  if  we  deliberately 
chose  to  weight  potassium  permanganate  as  a  billion 
times  as  important  as  every  other  commodity,  the  re- 
sulting index  number  for  the  1366  commodities  would 
praptically  coincide  with  the  price  movement  of  potas- 
sium  permanganate.  Likewise,  if  we  were,  instead,  to 
weight  oil  of  lemon  a  billion  times  as  important  as  every 
other,  the  index  number  would  become  practically  identi- 
cal therewith.  Obviously,  in  either  case,  we  should  be 
grossly  unfair.  In  the  one  case  our  index  number  would 
yield  the  absurd  conclusion  that  the  prices  of  1917  aver- 
aged 39  times  as  high  as  pre-war  prices.  In  the  other 
case  it  would  yield  the  equally  absurd  conclusion  that 
the  prices  of  1917  averaged  only  a  little  over  a  third  of 
the  pre-war  prices.  In  each  case  the  trouble  would  be 
that  a  commodity,  really  very  unimportant  as  compared 
to  wheat,  steel,  flour,  cotton,  and  hundreds  of  other  com- 
modities, would  be  treated  as  though  it  were  enormously 
re  important. 

We  are  not  yet  ready  to  say  what  system  of  weighting 
is  the  fairest,  nor  shall  we  be  ready  until  we  have  set  up 
certain  tests  of  fairness.  We  shall  then  reach  the  curious 
conclusion  that,  contrary  to  common  opinion,  no  system 
weighting  is  universally  the  fairest;  that  the  fairest 
weighting  for  the  arithmetic,  harmonic,  and  geometric 


UU 

of 


FOUR  METHODS  OF  WEIGHTING  45 

types,  for  instance,  are  all  different.  Here  we  must  be 
content  to  lay  the  foundations,  by  describing  the  four 
primary  systems  of  weighting  which  have  been  or  might 
be  set  up. 

As  we  have  seen,  weighting  any  term  in  an  index  num- 

r  is  virtually  counting  it  as  though  it  were  two  or  three, 
r  some  other  multiple,  as  compared  with  another  term 

unted  only  once.  This  applies  to  any  of  the  six  types 
of  averages.  / 

But  on  what  principle  shall  we  weight  the  terms? 
Arthur  Young's  guess  ancl  other  guesses  at  weighting 
represent,  consciously  or  unconsciously,  the  idea  that 
relative  money  values  of  the  various  commodities  should 
determine  their  weights.  A  value  is,  of  course,  the  prod- 
uct of  a  price  per  unit,  multiplied  by  the  number  of 
units  taken.  Such  values  afford  the  only  common  meas- 
ure for  comparing  the  streams  of  commodities  produced, 
exchanged,  or  consumed,  and  afford  almost  the  only 
basis  of  weighting  which  has  ever  been  seriously  proposed. 
If  sugar  is  marketed  to  the  extent  of  ten  billion  dollars' 
value  a  year,  while  salt  is  marketed  at  only  five  billion 
dollars'  value  a  year,  there  is  clearly  ground  for  regarding 
sugar  as  twice  as  important  as  salt. 


§  2.  Weighting  by  Base  Year  Values  or  by  Given 
\  Year  Values 

But  any  index  number  implies  two  dates,  and  the  values 
by  which  we  are  to  weight  the  price  ratios  for  those  two 
dates  will  themselves  be  different  at  the  two  dates. 

Constant  weighting  (the  same  weight  for  the  same  item 
in  different  years)  is,  therefore,  a  mere  makeshift,  never 
theoretically  correct,  and  not  even  practically  admissible 
when  values  change  widely.  In  Revolutionary  days, 


46  THE  MAKING  OF  INDEX  NUMBERS 

candles  were  important,  but  today  the  total  money  value 
of  the  candle  trade  is  negligible.  Rubber  tire  values  are 
important  today,  but  were  unimportant  two  decades 
ago.  In  comparing  the  price  levels  of  today  and  many 
years  ago  what  weight  shall  we  give  to  rubber  tires  or 
candles?  We  have  two  evident  choices.  We  may  take 
as  our  money  values  either  those  in  the  earlier  year  or 
those  in  the  later  year. 

§  3.  Numerical  Illustration 

j  It  often  makes  a  great  deal  of  difference  which  of  these 
Hwo  systems  of  weights  is  used.  Between  1913  and  1917 
some  commodities  rose  greatly,  not  only  in  price,  but  in 
money  value  marketed ;  others  scarcely  at  all.  In  gen- 
eral, the  36  commodities  rose  in  money  value  between 
1913  and  1917  about  100  per  cent  (their  total  value  rising 
from  $13,105,000,000  to  $25,191,000,000).  If  every  com- 
modity had  thus  doubled  in  money  value,  their  relative 
weights  would  remain  unchanged  so  that  it  would  make 
no  difference  to  the  index  number  which  year's  weights 
were  used. 

But,  as  a  matter  of  fact,  values  of  the  different  com- 
modities rose  very  unequally.  Some  rose  much  more 
and  others  much  less  than  100  per  cent.  Bituminous 
coal  had  a  value  of  $1.27  per  ton  X  477,000,000  tons,  or 
$606,000,000  in  1913,  and  in  1917,  $1,976,000,000,  or 
more  than  three  times  as  much.  Anthracite  coal,  on 
the  other  hand,  had  a  value  of  $35,000,000  in  1913,  and 
in  1917,  $44,000,000,  or  only  about  25  per  cent  more. 
Clearly,  then,  bituminous  coal  has  a  relatively  greater 
weight  when  the  1917  money  values  are  used  as  weights 
than  when  the  1913  values  are  used. 

Table  3,  assuming  1913  for  "base"  year  and  1917  for 
"given"  year,  shows  the  comparative  effect  of  weighting 


FOUR  METHODS  OF  WEIGHTING 


47 


according  to  1913  values  marketed  and  weighting  accord- 
ing to  1917  values  marketed.     The  figures  given  in  the 
Xtable  are,  of  course,  the  multipliers  to  be  used  in  weight- 
ing the  various  price  relatives. 


TABLE  3.    VALUES   (IN  MILLIONS  OF  DOLLARS)   OF 
CERTAIN  COMMODITIES 


COMMODITY 

1913 

(BASE  YEAR) 

1917 

(GIVEN  YEAR) 

Ktuminous  coal    

/     606 

1976 

s 

/       140 

604 

Pig  iron                  .        

462 

1502 

422 

1011 

Anthracite  coal                 

35 

44 

Petroleum  

1282 

1848 

Coffee 

96 

123 

Lumber  

1971 

2227 

All  of  the  first  four  items  in  the  table  are  examples  of 
commodities  the  prices  and  values  of  which  rose  extraor- 
dinarily from  1913  to  1917,  so  that  their  weights,  if  taken 
by  1917  figures,  would  be  very  great.  Contrariwise,  the 
last  four  items  are  examples  of  commodities  the  prices 
and  values  of  which  rose  very  little. 

A  glance  at  the  table  will  show  the  preponderance  in 
1913  of  the  last  four  commodities  taken  as  a  whole,  rela- 
tively to  the  first  four,  and  the  preponderance  in  1917  of 
the  first  four  relatively  to  the  last  four.  The  reason 
for  this  change  of  relative  weights  is,  of  course,  that  the 
upper  group  rose  more  in  price  than  the  lower. 

Let  us  calculate  an  index  number  by  both  the  two  con- 
trasted methods  of  weighting.  Let  the  type  of  index 
number  be,  say,  the  arithmetic  and  let  us  calculate  it 


48  THE  MAKING  OF  INDEX  NUMBERS 

for  1917  relatively  to  1913  as  a  base,  from  the  usual  data 
or  the  36  commodities.  We  begin  by  using  the  base 
values  as  weights.  This  kind  of  index  number  has 
as  its  formula,  No^-3.  For  bacon  in  1917,  the  price  rela- 
tive (as  previously  calculated)  isj.92.72  per  cent  and  the 
base  year  value  (12.36  cents  per  pound  X  1077  million 
pounds)  is  133.117  million  dollars.  For  barley,  the  price 
^relative  js  21 1.27  per  cent  and  the  base  year  value  (62.63 
cents  per  bushel  X  178.2  million  bushels)  is  111.607 
million  dollars^etc.  According  to  the  arithmetic  method, 
we  first  multiply  each  price  relative  by  its  weight  and 
then  divide  by  the  sum  of  the  weights.  The  results  are : 

For  bacon  192.72  per  cent  X  133JJ.7  million  dollars  =  256.54  million 
dollars. 

For  barley  211L27  per  cent  X  111.607  million  dollars  =  235.79  million 

dollars,  etc. 
( 

The  sum  of  all  36  such  results  is  21238.49  million  dol- 
lars, which  divided  by  the  sum  of  the  weights,  13104.818 
million  dollars,  gives  1.620£,  or  162.07  per  cent  as  the 
desiredjjndex  number.  ^ 

./  Thus,  the  arithmetic  index  number  for  1917,  when  the 
base  year  (1913)  system  of  weighting  is  used  (Formula 
3),  is 

A92.72X  f211.27\m  607   , 

V   100  r  '  \~100y1-  „  162.07 

13104.818  100 

In  other  words,  from  1913  to  1917,  the  price  level  rose 
(according  to  Formula  3)  from  100  to  162.07.  But,  by 
using  the  given  year  values  as  weights  (Formula  9), the 
resulting  index  number  is  180.72  per  cent,  exceeding  the 
former  (162.07)  by  11.51  per  cent. 

Likewise,  the  harmonic  index  number  for  1917, 
when  the  base  year  system  of  weighting  is  used  (Formula 
13),  is 


FOUR  METHODS  OF  WEIGHTING  49 

13104.818  147.19 


(  10°  \-i-i  mr-t-  =    10° 

l21L27;11L607  +  •  '  ' 

In  other  words,  from  1913  to  1917,  the  price  level  rose 
(according  to  Formula  13)  from  100  to  147.19.  But,  by 
using  the  given  year  values  as  weights  (Formula  19), 
the  resulting  index  number  is  161.05  per  cent,  exceeding 
the  former  (147.19)  by  9.42  per  cent. 

Likewise,    the    geometric    index    number,   for    1917, 
when  the  base  year  system  of  weighting  is  used  (Formula 

9^     i<a  U-      iWt.  UK 

do).  IS  BT  *-    *•' 


"^1/192.72^ /211.27\m.6Q7  _  154.08 

100  )         X  \~WO~)  100   ' 

vln  tther  words,  from  1913  to  1917,  the  price  level  rose 
(According  to  Formula  23)  from  100  to  154.08.     But  by 
ng  the  given  year  values  as  weights  (Formula  29),  the 
ulting  index  number  is  170.44,   exceeding  the  former 
54.08)  by  10.62  per  cent. 
Here  is  a  new  source  of  differences !     Not  only  does  it 
make  a  considerable  difference  what  type  of  average  is 
used,  —  wnether  arithmetic,   or  harmonic,  or  geometric, 
—  but  it  also  makes  a  great  difference  what  the  weighting 
is,  —  whether  base  year  weighting  or  given  year  weight-- 
ing,  or  simple  (i.e.  even)  weighting. 
^Were  we  to  stop  at  this  point,  we  should  be  even  more 
,^-krclined  to  join  N.  G.  Pierson  and  give  up  index  numbers 
~"  in  disgust  as  a  delusion  and  a  snare. 

§  4.   Graphic  and  Algebraic 

«r—  -^Graphically,  Chart  9  shows  the  contrast  between  the 
r^— two  weighted  arithmetic  index  numbers  as  well  as  the 
corresponding  contrasts  between  the  two  weighted  har- 
monic and  the  two  weighted  geometric  index  numbers. 
It  will  be  observed  that  the  upper  harmonic  (Formula 


50  THE  MAKING  OF  INDEX  NUMBERS 

19)  is  almost  coincident  with  the  lower  arithmetic  (For- 
mula 3),  the  other  arithmetic  and  harmonic  (Formulae 
9  and  13)  diverging  about  equally  on  opposite  sides  of 
these  central  lines.  The  two  geometries  (Formulae  23 
and  29)  lie  about  midway  —  one  between  the  two  har- 


The  Five -Tine  Fork 

of  6  Curves 

(Prices) 


\s* 


7J  74  75  16  '17  18 

CHART  9P.  Three  types  of  index  numbers,  the  arithmetic  (3  and  9), 
the  harmonic  (13  and  19),  and  the  geometric  (23  and  29),  each  type  being 
weighted  in  two  ways,  namely,  by  the  values  of  the  base  year  (3,  13,  23) 
and  by  the  values  of  the  given  year  (9,  19,  29),  forming  five  nearly  equidis- 
tant tines  of  the  fork.  In  each  case,  the  given  year  weighting  makes  for 
,  a  higher  position  of  the  curve  than  the  base  year  weighting.  (This  holds 
true  whether  prices  are  rising  or  falling.) 


monies  in  the  lower  half  of  the  chart  and  the  other  be- 
tween the  two  arithmetics  in  the  upper  half.  Each  of 
the  three  types  (arithmetic,  harmonic,  geometric)  thus 
has  its  two  curves  forking  about  equally;  but  their  re- 
spective forks  are  placed  in  three  substantially  equidis- 
tant positions,  the  lower  tine  of  the  uppermost  fork 
(arithmetic)  almost  coinciding  with  the  upper  tine  of  the 
lowermost  fork  (harmonic),  while  the  remaining  pair  of 
tines  (geometric)  split  the  other  two  pairs. 


FOUR  METHODS  OF  WEIGHTING  51 

Chart  10  shows  the  similar,  but  much  smaller,  contrast 
for  the  weighted  medians  (Formulae  33  and  39).  The 
"mode,"  were  it  charted,  would  show  even  less  contrast ; 
in  fact,  in  the  rough  approximation  here  used  it  shows 
none  at  all,  although,  strictly,  for  it  as  for  every  other 
type,  the  given  year  weighting  always  makes  for  a  higher 
index  number  than  does  the  base  year  weighting.1 


The  Five-Tine  Fork 

of  6  Curves 

(Quantifies) 


VJ  74  75  16  '17  IB 

CHART  9Q.   Analogous  to  Chart  9  P. 

V 

Algebraically,  the  arithmetic  index  number  weighted 
by  base  year  values  (Formula  3)  and  written  for  any  given 
year  (as  year  1)  relatively  to  the  base  year  (year  0)»  is 
evidently 


+  p'o  q'o  +   •  •  • 
or,  by  the  shorter  method  of  writing, 


In  like  manner,  the  arithmetic  index  number  weighted 
by  given  year  values  (Formula  9)  is 


1  As  to  calculating  the  weighted  median  and  mode  see  Appendix  I 
(Note  to  Chapter  III,  §  4). 


52          THE  MAKING  OF  INDEX  NUMBERS 


The  weighted  formulae  for  other  types  are  given  in  Ap- 
pendix V. 

§  5.  Weighting  by  Base  Year  Values  Easiest 

The  weighting  by  base  year  values  has  been  employed 
by  statisticians  more  frequently  than  the  weighting  by 

The  Two  Extreme  Methods  o? 

Weighting  Median 
(Prices) 


15  W  15  16  17  78 

CHART  10P.  Showing  the  median  type  of  index  number,  weighted  by 
the  values  of  the  base  year  (33)  and  by  the  values  of  the  given  year  (39), 
the  latter  weighting  resulting,  as  before,  in  a  higher  curve  than  the  former. 
The  difference  between  the  two  weightings  is  not  so  great  as  in  the  case  of 
the  arithmetic,  harmonic,  and  geometric  types,  indicating  that  the  matter 
of  weighting  makes  less  difference  to  the  median  than  it  does  to  those  types. 

given  year  values  because,  with  a  fixed  base,  only  one  set 
of  values  needs  to  be  worked  out  for  a  whole  series  of  index 
numbers.  Calculating  only  one  set  of  values  saves  labor 
as  compared  with  calculating  a  separate  set  for  each  given 
year.  Another  reason  why  weighting  by  base  values 
has  so  often  been  employed  is  that  often  only  one  set  of 
weights  can  be  worked  out.  For  instance,  a  census  year 


FOUR  METHODS  OF  WEIGHTING  53 

may  give  the  data  required  for  starting  off  an  index  num- 
ber with  that  census  year  as  a  base  while  similar  data  for 
the  succeeding  years  may  be  unavailable  for  want  of  a 
yearly  census. 

The  United  States  Bureau  of  Labor  Statistics  has  used 
base  weighting  with  an  arithmetic  type  of  index  number. 
The  Harvard  Committee  on  Economic  Research,  in  the 
Day  index  number  of  production,  employs  it  with  a  geo- 
metric type.  Weighting  by  given  year  values  (as  in 
Formula  9)  has  been  proposed  by  Palgrave  for  arithmetic 
index  numbers. 

The  Two  Extreme  Methods  of 

Weighting  Median 
(Quantities] 


yj  14  15  76  77  18 

CHART  10Q.  Analogous  to  Chart  10P. 

§  6.  Two  Intermediate  Systems  of  Weighting 

Besides  the  two  systems  of  weighting  which  have  just 
been  described  there  are  two  other  analogous  systems, 
making  four  in  all.  Of  these  four,  the  system  of  base 
value  weights  will  be  called  "  weighting  /"  and  the  sys- 
tem of  given  value  weights  will  be  called  "  weigh  ting  7F." 
The  other  two  systems  (II  and  777)  still  to  be  described 
fall  logically  between  these  extremes.  In  Systems  77 
and  777  each  commodity  is  weighted  by  a  hybrid  value, 
relating  not  to  the  base  year  alone  nor  the  given  year 
alone  but  partly  to  one  and  partly  to  the  other.  In  sys- 


54  THE  MAKING  OF  INDEX  NUMBERS 

tern  II  the  value  is  made  by  multiplying  the  price  of  each 
commodity  in  the  base  year  by  the  quantity  of  that  com- 
modity in  the  given  year.  In  system  III  each  commodity 
is  weighted  by  the  other  hybrid  value  formed  by  multi- 
plying its  price  in  the  given  year  by  its  quantity  in  the 
base  year.  That  is : 

/,  each  weight  =  base   year  price  X  base  year  quantity 
weight  —  base  year  price  X  given  year  quantity 
each  weight  =  given  year  price  X  base  year  quantity 
each  weight  =  given  year  price  X  given  year  quantity 

Algebraically,  the  weights  used  in  the  four  systems  of 
arB,  respectively : 

/.   paqo,  p'off'o,  etc. 
II.   poqi,  p'oq'i,  etc. 

0,  p'iq'o,  etc. 

1,  p'iq\,  etc. 

In  the  following  Table  4,  of  weights,  if  we  take  the  same 
eight  commodities  previously  cited  (§3  above)  and  apply 
the  weight  systems  II  and  777,  we  find  that,  while  every 
figure  has  changed,  there  are  still  the  same  marked  tend- 
encies as  in  Table  3.  In  the  first  column,  the  lower  group 
of  four  articles  preponderates  over  the  upper  group  of  four 
articles ;  and  in  the  second  column  vice  versa. 

Thus,  in  both  tables,  the  relative  importance  of  the 
two  groups  of  commodities  changes  greatly  between  one 
column  and  the  other.  The  reason  is  that  the  two  groups 
of  commodities  are  purposely  contrasted  as  to  price 
change  (but  not  as  to  quantity  change).  It  follows  that, 
if  the  second  column  weights  are  used,  the  upper  four 
commodities  which  rise  the  more  in  price  will  be  the 
more  heavily  weighted,  while  the  opposite  is  true  if  the 
first  column  weights  are  used.  These  points  will  be 
elaborated  in  Chapter  V. 


FOUR  METHODS  OF  WEIGHTING 


55 


TABLE  4.    HYBRID  VALUES  (IN  MILLIONS  OF  DOLLARS)  OF 
CERTAIN  COMMODITIES 


COMMODITY 

1913  PRICES  MULTIPLIED 
BT  1917  QUANTITIES 

1917  PRICES  MULTIPLIED 
BY  1913  QUANTITIES 

Bituminous  coal    .  . 
Coke   

701 
172 

1708 
494 

Pig  iron 

577 

1203 

Oats  

596 

715 

Anthracite  coal  .... 
Petroleum  

40 
1835 

39 
1292 

Coffee   .           .   . 

147 

80 

Lumber      

1916 

2290 

Let  us  trace  specifically  the  effects  of  all  four  systems 
of  weighting. 

The  arithmetic  formulae  give  the  following  index  num- 
bers for  1917,  relatively  to  1913  as  the  base : 

Arithmetic  by  weight  system     /  (Formula  3)  162.07  per  cent 

II  (      "        5)  161.05  per  cent 
"       "  "      ///  (       "        7)  180.53  per  cent 

"       "  "       IV  (      "        9)  180.72  per  cent 

The  harmonic  f ormulse  give : 


Harmonic  by  weight  system     7  (Formula  13)  147.19  per  cent 
11          "       "  "        II  (      "        15)  144.97  per  cent 

"          "       "  "      ///  (       "        17)  162.07  per  cent 

"          "       "          "       IV  (      "        19)  161.05  per  cent 


The  geometric  f  ormulse  give : 

Geometric  by  weight  system     /  (Formula  23)  154.08  per  cent 

II  (      "        25)  152.45  per  cent 

"  "       "  "      ///  (       "        27)  170.82  per  cent 

"          "       "          "       IV  (      "        29)  170.44  per  cent 

Numerically,  the  above  calculations  show  that  weight 
system  II  gives  results  almost  identical  with  weight  sys- 
tem /,  while  likewise  weight  systems  777  and  IV  are  al- 


56  THE  MAKING  OF  INDEX  NUMBERS 

most  identical,  there  being  a  wide  gap,  however,  between 
these  two  pairs. 

This  disparity,  as  indicated,  is  due  to  the  fact  that,  in 
deriving  the  weights  I  and  77,  base  year  prices  are  used, 
/while  in  777  and  7F,  given  year  prices  are  used,  the  prices 
Jin  both  cases  out-influencing  the  quantities. 

The  same  contrasts  (of  7,  77  as  against  777,  IV),  though 
less  pronounced,  are  found  in  the  weighted  medians; 
but  in  the  modes  these  contrasts,  while  present,  are  im- 
perceptible. 

We  have  now  cited  not  only  a  simple  arithmetic  for- 
mula, but  four  weighted  arithmetic  formulae,  and  like- 
wise a  simple  and  four  weighted  harmonic  formulae,  a 
simple  and  four  weighted  geometric  formulae,  a  simple 
and  four  weighted  median  formulae,  and  a  simple  and  four 
weighted  mode  formulae,  for  obtaining  index  numbers. 

§  7.  Only  Two  Systems  of  Weighting  far  tfrp  Aggregating 

Up  to  this  point,  therefore,  we  have  considered  foui 
forms  of  weighting  for  each  of  the  five  types  of  index 
numbers.  The  sixth,,  or  aggregative,  type  of  index  num- 
ber has  as  yet  been  considered  only  in.  its  " simple"  form. 
Because  of  its  peculiar  construction  it  is  capable  of  only 
two  systems  of  weighting  at  all  analogous  to  those  we  have 
belen  considering.  As_jwe  have  seen,  the  simple  aggre- 
gative is  a  very  peculiar  average  of  pric,e  ratios  (price 
relatives)  being"a7atio  oTthesums  of  the  prices  themselves. 
Thus  the  simple  aggregative  gives : 

sum  of  all  prices  of  1917  __  index  number  for  1917  rela- 
sum  of  all  prices  of  1913  "       tively  to  1913. 
Consequently  the  weighting  cannot  be  applied  to  the 
price  ratios  as  such,  but  must  be  applied  directly  to  the 
pSces  themselves  —  fceth  in  numerator  and  denominator. 

— --= 


FOUR  METHODS  OF  WEIGHTING  57 

Of  course,  the  same  weight  is  to  be  applied,  in  this  way, 
to  the  prices  of  the  same  items  in  both  numerator  and 
denominator. 

Now,  in  the  previous  formulae,  the  weights  were  values. 
But  value  is  price  multiplied  by  quantity.     In  the  aggre- 
//gative  formulae,  however,  the  price  part  is  already  there 
the  only  thing  to  be  weighted. 

It    would    be    absurd    to    multiply    price    by   value 
(which  already  contains  price).     Consequently,   in  the 
,  aggregative  formulae,  the  weights  must  be  just  quantities 
^and  these  quantities  must  be  either  the  quantities  of  the 
/base  year  (1913)  or  the  quantities  of  the  given  year  (1917). 
Xlf  we  wish  to  keep  up  the  analogy  with  the  four  kinds  of 
Weighting,  used  for  all  the  other  types,  we  may  consider 
the  weighting  of  the  aggregative  by  base  year  quantities 
/^,s  weighting  I  (Formula  53),  and  the  weighting  by  given 
/year  quantities  as  weighting  IV  (Formula  59),  omitting 
/  II  and  III  entirely.1 

§  8.  Numerical  Calculation  of  Weighted  Aggregative 

Numerically,  to  illustrate  by  our  36  commodities,  let 
us  outline  the  calculation  of  the  aggregative  by  base 
weighting  (Formula  53)  for  the  index  number  of  1914 
(relatively  to  1913  as  base).2  This  is  defined  as  the  ratio 
of  the  sum  of  the  hybrid  values  for  1914  (because  reckoned 
with  the  quantities  of  1913)  to  the  true  values  for  1913. 

The  denominator  of  this  fraction,  i.e.  the  true  value 
in  1913,  is,  as  above,  $13,104,818,000.  The  numerator 
is  derived  in  a  similar  way.  Beginning  with  bacon,  we 
obtain  its  (hybrid)  value  by  multiplying  its  price  in  1914 
(12.95  cents  per  pound)  by  its  quantity,  not  in  1914  but 

1  See  Appendix  I  (Note  to  Chapter" III,  §  7). 

8  For  model  examples  to  aid  in  the  practical  calculation  of  this  as  well  as 
of  eight  other  sorts  of  index  numbers,  see  Appendix  VI,  §  2. 


58  THE  MAKING  OF  INDEX  NUMBERS 

in  1913  (1077  million  pounds),  obtaining  $0.1295  X  1077, 
or  139.47  million  dollars.  Similarly,  the  barley  value  is 
62.04  cents  per  bushel  X  178.2  million  bushels,  or 
110.56  million  dollars,  and  so  on,  the  total  of  the  36 
jguch  values  being  $13,095,780,000,  the  desired  numerator. 

The  ratio  of  this  numerator  to  the  above  denominator 
comes  out  99.93  per  cent,  the  index  number  sought.  This 
is  by  Formula  53,  weighting  7. 

.For  Formula  59,  using  " given  year"  weighting  77,  the 
y^alculation  is  similar.1  The  numerator  is  13033.034,  the 
sum  of  the  true  values  in  1914,  and  the  denominator  is 
12991.81,  the  sum  of  the  hybrid  values  for  1913  (found  by 
using  the  prices  of  1913  and  the  quantities  of  1914).  The 
ratio  of  the  numerator  to  the  denominator  is  100.32  per 
cent,  almost  the  same  as  the  99.93  per  cent  by  the  other 
formula  (53). 

I  These  two  index  numbers  (Formulae  53  and  59),  con- 
trasted merely  as  to  whether  base  year  quantities  or 
given  year  quantities^are  used,  show  no  tendency  to  the 

I/  wide  contrast  between  base  year  and  given  year  weighting 

l/lound  in  the  arithmetic,  harmonic,  and  geometric  index 

numbers.     There  is  no  tendency  for  Formula  53,  in  which 

base  year  quantities  are  used,  to  be  less  than  Formula  59, 

in  which  given  year  quantities  are  used.    The  two  curves 

, are  very  close  together  and  even  cross  each  other.    As 

/    1^  the  reader  may  suspect,  the  reason  for  this  close  similarity 
*      II  is  that  the  price  element  which,  in  previous  weighting 

II  systems  was  the  disturbing  element,  is  here  missing,  the 
({weights  being  mere  quantities. 

§  9.  The  Algebraic  Formulae 

Algebraically,  the  aggregative  index  number  for  prices 
with  base  year  weighting  or  weighting  7  (Formula  53)  is 

1  See  also  Appendix  VI,  §  2. 


FOUR  METHODS  OF  WEIGHTING  59 


+  p'lq'o  +  p"ig"o  +   •  .  . 

Poffo  +  p'o  ff'o  +  p'Vo  +  '.  .  . 
or, 

J 


while  with  the  given  year  weighting,  or  weighting  IV,  the 
aggregative  (Formula  59)  is 


The    corresponding    index    numbers    of    quantities 
(weighted  by  prices)  are 


/   (base  year  prices)         ?, 
v—  *  —  J  — 

IV  (given    " 


§  10.  Historical 

The  first  of  the  two  weighted  aggregative  formulae, 
°    (Formula  53  for  prices),  is  the  form  used  by  the 


United  States  Bureau  of  Labor  Statistics.  It  is  a 
return  to  an  old  idea,  since  this  method  was  explicitly 
formulated  and  advocated  by  Laspeyres  in  1864,  and 
Walsh  gives  it  the  name  of  Laspeyres  '  method.1  ^^^^ 
The  present  vogue  of  this  method  is  largely  due  to  the 
vigorous  advocacy  of  it  and  strong  arguments  for  it  made 
by  G.  H.  Knibbs,  the  Government  Statistician  for  Aus- 
tralia. It  has  been  formally  recommended  by  a  vote  of  a 
recent  conference  of  the  statisticians  of  the  British  Em- 
pire. 

The  second  of  the  two  formulae,  ^Ml  (59)     was   ad- 


Se^e  Walsh,  The  Measurement  of  General  Exchange  Value,  p.  558. 


60  THE  MAKING  OF  INDEX  NUMBERS 

vocated  and  employed  by  Paasche  in  1874.    Walsh  calls 
it  Paasche's  method.1 

These  two  names  will  recur:    Laspeyres'  formula,  53 
(aggregative    weighted   7)    and   Paasche's   formula,    59 
-  (aggregative  weighted  IV). 

§  11.  Relation  of  Weighted  Aggregative  to  Weighted 
Arithmetic  and  Weighted  Harmonic 

It  is  of  much  interest  to  note  that  the  arithmetic  aver- 
age weighted  by  basBJ^ah^s  (7^  or  Formula  3)  necessarily 
reduces,  when  simplified,  to  Laspeyres'  formula  (53)  — 
that  is,  the  aggregative  average  weighted  by  base  quanti- 
ties ;  while  the  harmonic  average  weighted  by  given  year 
values  (IV y  or  Formula  19),  when  simplified,  likewise 
reduces  to  Paasche's  formula  (59)  —  that  is,  the  aggre- 
.  gative  average,  weighted  by  given  year  quantities ;  and 
^hat  furthermore  the  arithmetic  average  weighted  by 
weight  method  77  (Formula  5)  reduces  to  Paasche's; 
and  the  harmonic  average  weighted  by  weight  method 
777  (Formula  17)  reduces  to  Laspeyres'.2 

Algebraically,  the  proof  of  these  propositions  is  simple.8 
Graphically,  from  what  has  been  said  it  follows  that 
each  of  the  two  central  curves  of  Chart  9  has  a  triple 
meaning.  Each  represents  an  arithmetic,  harmonic, 
and  aggregative  index  number.  What  is  labeled  3  might 
be  labeled  also  17  and  53  and  what  is  labeled  19  might 
be  labeled  also  5  and  59. 

§  12.  Formulae  thus  far  Available 

We  see,  then,  that  there  are  four  primary  methods  of 
weighting  (7,  77,  777,  77)  applicable  to  five  of  the  six 

1  The  Measurement  of  General  Exchange  Value,  p.  559. 

*  Ibid.,  pp.  306-7,  350,  352,  511.  Walsh  was  the  first  to  point  out  these 
identities  excepting  that  which  he  refers  to  as  having  been  first  pointed  out 
by  me  (i.e.  3  and  53).  *  gee  Appendix  I  (Note  to  Chapter  III,  §  11). 


FOUR  METHODS  OF  WEIGHTING 


61 


types  of  index  numbers,  namely,  arithmetic,  harmonic, 
geometric,  median,  mode,  and  two  analogous  methods 
(/,  IV)  applicable  to  the  sixth  (aggregative).  Let  us 
now  "take  account  of  stock"  and  see  what  index  num- 
bers we  have  thus  far  obtained  all  together.  We  have 
the  following : 

TABLE    5.    IDENTIFICATION    NUMBERS    OF    PRIMARY 

FORMULA 


WEIGHTING 

ARITH. 

HARM. 

GEOM. 

MEDIAN 

MODE 

AGGREG. 

Simple  (or  even)  . 
7.   Base  year  only  .  .  . 
II.   Base  year  prices  X 
given  year  quan- 
tities 

1 
& 

jS} 

11 
13 

15 

21 
23 

25 

31 
33 

35 

41 
43 

45 

51 

///.   Given  year  prices  X 
base  year  quan- 
tities. .        ... 

159 

7 

27 

37 

47 

IV.  Given  year  only  .  . 

9 

& 

29 

39 

49 

@ 

This  variety  may  seem  at  first  merely  to  increase  our 
sense  of  bewilderment  and  distrust  of  index  numbers. 
But  we  shall  find  grounds  for  discriminating  between  the 
various  formulae.  Moreover,  as  has  been  noted,  and  as 
is  evident  from  inspecting  the  formulae  in  Appendix  V, 
there  are  four  duplications  in  the  table  (53  =  3  =  17; 
59  =  19  =  5). 

These  various  weighting  systems  are,  of  course,  not 
the  only  possible  ones.  In  Chapter  VIII  we  shall  consider 
systems  formed  by  taking  averages  or  means  of  the  above 
varieties  of  weights.  There  seem  to  be  no  others  pro- 
posed worth  very  serious  attention.1 

1  See  Appendix  I  (Note  to  Chapter  III,  §  12). 


CHAPTER  IV 
TWO  GREAT  REVERSAL  TESTS 

§  1.  Reversal  Tests  in  General 

As  indicated  at  the  close  of  the  last  chapter,  not  all 

index  numbers  have  equal  claims  to  be  considered  as 

^  truly  representative  of  price  movements.    They  may  be 

good,  bad,  or  indifferent,  and  our  next  task  is  to  set  up 

--certain  criteria  for  distinguishing  them  as  such. 

T     The  fundamental  question,  mentioned  in  Chapter  I, 

r*-§  6,  is  that  of  fairness.    The  requirement  of  fairness  is 

^often  exp^ggged  by  the  demand,  "  put  yourself  in  his  place." 

^  Fairness  is  not  fair  which  takes  account  of  whose  ox  is 

_^  gored.     In  short,  "  It  is  a  poor  rule  that  won't  work  both 

^ways."     This  kind  of  test,   "the  golden  rule"   of  fair 

*•  dealing  among  men  is,  in  a  sense,  the  golden  rule  in  the 

«  domain  of  index  numbers  also. 

Index  numbers  to  f>e  fair  ought  to  work  both  ways  — 
-both  ways  as  regards  an^y  tw,o  commodities  to  be  averaged, 
or  as  regards  the  two-times  to  be  compared,  or  as  regards 
the  two  sets  of  associated  elements  for  which  index 
numbers  may  be  calculated  —  that  is,  prices  and  quan- 
tities. The  rule  of  changing  places  applies  separately 
*  to  each  of  the  three  following  sets  of  magnitudes : 
/irst/the  several  commodities;  second,  the  two  times; 
third,  the  two  factors  —  prices  and  quantities.  To  be 
specific,  this  rule  of  changing  places  means  three  separate 
things:  interchanging  any  two  commodities,  inter- 
changing the  two  times,  interchanging  prices  and 


TWO  GREAT  REVERSAL  TESTS  63 

quantities.  In  short,  we  must,  in  some  sense,  treat 
alike :  (a)  any  two  commodities ;  (6)  the  two  times ; 
(c)  the  two  factors. 

The  first  test  is  seldom  if  ever  violated.     It  is  rnen- 
tioneoThere  for  completeness  and  to  afford  a  basis  for  a  bet- 
ter appreciation  of  the  two  less  obvious  tests  which  follow. 
-  In  order  to  avoid  confusion  the  three  tests  will  be  dis- 
tinguished as: 

^Preliminary"  —  The  commodity  reversal  test 
'  Test  1  —  The  time  reversal  test 

-"Test  2  —  The  factor  reversal  test 

Any  formula  to  be  fair  should  satisfy  all  three  tests. 
JThe  requirement  as  to  commodities  is  that  the  order  of  the 
commodities  ought  to  make  no  difference  —  that,  to  be 
specific,  any  two  commodities  could  be  interchanged,  i.e. 
their  ^jder  reversed,  without  affecting  the  resulting  index 
number.     This  is  so  simple  as  never  to  have  been  for- 
mulated.    It  is  merely  taken  for  granted  and  observed 
instinctively.    Any  rule  for  averaging  the  commodities 
must    be    so  general   as .  to   apply  interchangeably    to 
all  of  the  terms  averaged.     It  would  not  be  fair,  for 
instance,  arbitrarily  to  average  the  first  half  of  the  com- 
modities by  the  arithmetic  method  and  the  other  half 
by  the  geometric,  nor  fancifully  to  weight  the  seventh 
'    commodity  by  7  and  the  tenth  commodity  by  10  so  that 
xif  the  seventh  and  tenth  commodities  were  interchanged 
^  the  result  would  be  affected.1 

1  It  may  be  worth  while,  for  contrast,  to  note  an  example  of  an  average, 
in  another  field  of  thought,  for  which  the  order  of  the  terms  is  not  inter- 
changeable. If  the  German  Reparation  Debt  were  represented  by  bonds 
of  100  billion  marks  drawing  10  per  cent  interest  for  the  first  15  years, 
6  per  cent  for  the  next  15  years,  and  3  per  cent  for  a  third  period  of  15  years, 
the  "average"  rate  of  interest  for  all  three  periods  will  not  be  independ- 
ent of  the  order.  It  would  be  different  if,  for  instance,  the  first  period 
were  at  3  per  cent  and  the  last  at  10  per  cent.  (See  Irving  Fisher's  T^z 
Rate  of  Interest,  New  York,  1907,  p.  372.) 


64  THE  MAKING  OF  INDEX  NUMBERS 

The  other  two  tests  mentioned  (which  will  be  referred 

to  as  Test  1  and  Test  2),  although  thoroughly  analogous 

,     to  the  Preliminary  Test,  have  not  been  so  well  observed. 

s~   On  the  contrary  many  index  numbers  in  actual  use  fail 

to  observe  either  of  them,  and  none  at  all  observe  the 

^  second ! 

§  2.  The  Time  Reversal  Test  * 

Just  as  the  very  idea  of  an  index  number  implies  a  set 
of  commodities,  so  it  implies  two  (and  only  two)  times 
(or  places).  Either  one  of  the  two  times  may  be  taken 
as  the  " base."  Will  it  make  a  difference  which  is  chosen  ? 
Certainly  it  ought  not  and  our  Test  1  demands  that  it 
shall  not.  More  fully  expressed,  the  test  is  that  the  for- 
mula for  calculating  an  index  number  should  be  such  that 
it  will  give  the  same  ratio  between  one  point  of  comparison 
and  the  other  point,  no  matter  which  of  the  two  zs  taken  as 
the  base. 

Or,  putting  it  another  way,  the  index  number  reckoned 
forward  should  be  the  reciprocal  of  that  reckoned  back- 
ward. Thus,  if  taking  1913  as  a  base  and  going  forward 
to  1918,  we  find  that,  on  the  average,  prices  have  doubled, 
then,  by  proceeding  in  the  reverse  direction,  we  ought 
to  find  the  1913  price  level  to  be  half  that  of  1918,  from 
which  we  started  as  a  base.  Putting  it  in  still  another 
1  way,  more  useful  for  practical  purposes,  the  forward  and 
backward  index  number  multiplied  together  should  give 
unity. 

The  justification  for  making  this  rule  is  twofold  :  (1)  no 
Reason  can  be  assigned  for  choosing  to  reckon  in  one 
-  direction  which  does  not  also  apply  to  the  opposite,  and 
(2)  such  reversibility  does  apply  to  any  individual  com- 
modity.    If  sugar  costs  twice^as  much  in  1918  as  in  1913, 
tLen  necessarily  it  costs  half  as  much  in  1913  as  in  1918. 


TWO   GREAT  REVERSAL  TESTS  65 

By  analogy  we  demand  that  any  formula  for  an  index 
/   number,  by  which  we  find  the  price  level  of  1918  is  double 
that  of  1913,  ought  to  tell  us  that  the  price  level  of  1913 
is  half  that  of  1918. 

This  requirement  is  still  more  appealing  to  our  sense 
~7>f  fairness  if  we  take  not  two  times,  but  two  places ;  we 
might  be  confused  by  the  fact  that  succession  in  time  is 
'different,  forward  from  backward,  and  wonder  for  a 
moment  whether  there  might  not  be  some  hidden  but  logi- 
cal reason  for  using  the  earlier  of  the  two  dates  as  the  base 
rather  than  the  later.  But  in  comparisons  between  places 
there  is  not  even  this  semblance  of  a  reason  for  regarding 
one  of  the  two  points  of  comparison  as  the  base  rather 
than  the  other. 

§  3.  The  Time  Reversal  Test  Illustrated  Numerically 

Yet  most  forms  of  index  numbers  in  use  do  not  con-  v 
form  to  this  reversal  test !    For  instance,  the  simple  arith- 
metic average  does  not. 

Numerically,  the  following  illustrations  show  this. 
Suppose  the  price  of  bread  is  twice  as  high  in  Philadelphia 
as  in  New  York  (20  cents  a  loaf  as  against  10  cents)  and, 
reversely,  the  price  of  butter  is  twice  as  high  in  New  York 
as  Philadelphia  (60  cents  a  pound  instead  of  30  cents). 
In  price  relatives  or  percentages,  taking  New  York  prices 
as  100  per  cent,  the  figures  are : 

Bread :    New  York  100  per  cent        Philadelphia  200  per  cent 
Butter:      "        "     100    "      "  "  50    "      " 

The  simple  arithmetic  index  number  for  Philadelphia 

is — — ,  or  125  per  cent,  and  would  make  it  appear 

2t 

that  bread  and  butter  were  on  the  average  25  per  cent 
higher  in  Philadelphia  than  in  New  York.  But  if  we 
take  Philadelphia  as  100  per  cent,  the  figures  are : 


66  THE  MAKING  OF  INDEX  NUMBERS 

Bread :    Philadelphia  100  per  cent        New  York    50  per  cent 
Butter:  "  100   "      "  "        "     200    "      " 

This  gives  5Q+  2QQ  =  125  per  cent,  or  25  per  cent  higher 
2i 

in  New  York  than  in  Philadelphia.  Since  each  city  can- 
not be  25  per  cent  above  the  other,  something  must  be 
wrong  with  the  formula  which  yields  such  a  preposterous 
result.  No  reason  can  be  assigned  why  the  formula  should 
be  applied  with  New  York  as  a  base,  which  will  not  equally 
justify  making  Philadelphia  the  base ;  and  no  more  reason 
can  be  assigned  for  making  one  of  any  two  years  compared 
the  base  which  will  not  equally  justify  making  the  other 
year  the  base. 

Again,  suppose  bread  rose  in  price  between  1913  and 
1918  from  10  cents  to  15  cents  a  loaf,  i.e.  in  price  relatives 
or  percentages  from  100  to  150,  and  butter  from  20  cents 
per  pound  to  50  cents  per  pound,  or  from  1QQ  tq_250.     The* 
index  number  forJ^lS  relatively  to  1913  as  a  base  is  then 

— —      ^_  .=•  200  per  cent.     But,  reversing  the  comparison 

2i 

and  taking  1918  as  the  base,  we  find  the  price  ratios  for 
1913jto  be,  for  bread,  66f  per  cent  and,  for  butter,  40  per 
cent,  The  average  of  these  is  not  the  required  50  per 
cent  but  53|  per  cent.  Consequently,  the  product  of 
the  two  opposite  index  numbers  is  not,  as  it  should  be, 
unity,  or  100  per  cent,  but  200  X  53£  =  106f  per  cent, 
or  6f  per  cent  too  great.  ^ 

Again,  taking  the  simple  arithmetic  average  of  the 
36  price  relatives  for  1917  relatively  to  1913,  or  175.79 
per  cent,  and  reversely,  taking  the  simple  arithmetic 
average  of  the  same  prices  for  1913  relatively  to  1917, 
or  63.34  per  cent,  and  multiplying  these  two  together 
we  get,  not  unity  or  100  per  cent,  but  111.35  per  cent. 
Evidently  there  is  here  an  error  of  11.35  per  cent. 


TWO  GREAT  REVERSAL  TESTS  67 

That  is,  the  simple  arithmetic  average,  checked  up  by 
itself  forward  and  backward  in  time,  stultifies  itself  by 
exactly  11.35  per  cent.  The  error  of  11.35  per  cent 
must  rest  somewhere.  It  may  be  that  the  175.79  per 
cent  for  1917  relatively  to  1913  is  too  high  by  11.35  per 
cent,  or  it  may  be  that  the  63.34  per  cent  for  1913  rela- 
tively to  1917  is  too  high  by  11.35  per  cent,  or  it  may  be 
that  the  two  figures  (175.79  and  63.34)  share  the  error, 
equally  or  unequally.  We  cannot  say.  What  we  can  say 
is  that  both  the  175.79  and  the  63.34  cannot  be  true  at 
once  and  that  between  them  there  is  a  total,  or  net  joint 
error  of  exactly  11.35  per  cent. 

Again,  we  find  that  the  simple  arithmetic  index  number 

of  the  36  commodities  makes  out  the  price  level  of  1915 

15     to  be  If  per  cent  higher  than  that  of  1914  with  1914  as 

xthe  base  while,  reversely,  it  makes  out  the  price  level  of 

'  1914  to  be  i  per  cent  higher  than  that  of  1915  with  1915 

as  the  base.     In  other  words  here  is  an  actual  case  where 

each  of  two  years  is  represented  by  the  arithmetic  index 

number  as  being  higher  priced  than  the  other ! 

The  simple  harmonic  index  number  also  fails  to  meet 
Test   1. 

The  simple  geometric  index  number,  on  the  other  hand, 
conforms  to  Test  1.     It  gives  166.65  per  cent  for  1917 
relatively  to  1913  and  60.01  per  cent  for  1913  relatively 
to  1917,  the  product  of  which  is  exactly  100  per  cent. 
The  general  proof  of  this  is  deferred  to  Chapter  VI. 
^    This  conformity  to  Test  1  (time  reversal)  does  not, 
^^crf  course,  prove  that  the   geometric  index  number  is 
^exactly  correct.     It  means  simply  what  it  says,  that  the 
r^simple  geometric  is  self-consistent  when  applied  reversely 
'"'in  time.     There  may  be  errors  in  both  figures  which  offset 
other  when  they  are  multiplied  but  there  is  no  net 
joint  error  in  the  product.    All  we  can  say  is  that  we 


68 


THE  MAKING  OF  INDEX  NUMBERS 


know  the  simple  arithmetic  index  number,  for  instance, 
has  failed  to  tell  the  truth  and  that  we  have  not  yet  caught 

Forward  (17-18)  and 
Backward  (16-17) 

Simple  Arithmetics 
contrasted 
(Prices) 

•17  78 


77 


CHART  HP.  Each  line  forward,  representing  the  changing  price  of  a 
commodity  between  1917  and  1918,  is  prolonged  backward  to  represent  the 
reciprocal  change  from  1918  to  1917.  Yet  the  simple  arithmetic  averages 
of  these  two  fans  of  lines  are  not  prolongations  of  each  other. 

the  simple  geometric  in  a  lie.    We  must  wait  till  we  apply 
to  it  Test  2. 

The  simple  median,  mode,  and  aggregative  all  fulfill 
Test  1.  The  general  proof  is  deferred  to  Chapter  VI. 


TWO  GREAT  REVERSAL  TESTS 


69 


Forward  <77-7<3W 
Backward (16-17) 

Simple   Arithmetics 
contrasted 

(Quantities) 


arithmetic 


CHART  11Q.  Analogous  to  Chart  IIP. 


70  THE  MAKING  OF  INDEX  NUMBERS 

But  none  of  the  weighted  index  numbers  yet  described 
conforms  to  Test  1.  Thus,  only  four  out  of  the  28  kinds 
of  index  numbers  so  far  encountered  fulfill  Test  1. 

§  4.  The  Time  Reversal  Test  Illustrated  Graphically 

We  have  seen  that  an  index  number  calculated  forward 
should  be  the  reciprocal  of  the  index  number  calculated 
backward.  Such  harmonious  results  would  be  repre- 
sented by  parallel  lines  in  our  charts.  But  in  the  case 
of  the  arithmetic  average  the  two  lines  will  not  be  parallel  ; 
that  is,  the  arithmetic  backward  is  not  the  reciprocal  of 
the  arithmetic  forward. 

Graphically,  this  is  illustrated  in  Charts  IIP  and  11 Q 
which  repeat  from  Charts  3P  and  3Q  the  dispersion  of 
the  36  individual  prices  (and  the  36  quantities)  from  1917 
to  1918.  To  represent  the  reverse  dispersion  from  1918 
to  1917,  in  order  not  to  let  the  two  sets  of  radiating  lines 
interfere  with  one  another,  and,  for  simplicity,  they  have 
been  radiated  from  the  same  point,  simply  to  the  left  in- 
stead of  to  the  right.  We  thus  really  have  two  separate 
charts ;  the  one  common  point  representing  1917  for  the 
right  hand  chart  but  representing  1918  for  the  left  hand 

chart.  iv\dLcJtA-t*£ 

N*w  the  line  for  ^Jri$f  mrnvmuSl  commodity  drawn 
backward  must,  in  our  ratio  method  of  charting,  take  the 
same  direction  as  that  for  the  same  commodity  drawn 
forward,  so  that  the  left  set  of  radiating  lines  are  simply 
the  backward  prolongations  of  the  right  set. 

But  (and  this  is  the  point  to  be  noted)  while  each  of 
these  36  price  lines  individually  is  the  prolongation  of 
its  mate,  yet  the  two  opposite  lines  for  their  average  (the 
arithmetic  index  number)  are  not  the  prolongations  each 
of  the  other.  The  two  longer  and  darker  lines  represent 
these  arithmetic  index  numbers  forward  and  backward ; 


TWO  GREAT  REVERSAL  TESTS  71 

and,  while  the  arithmetic  forward  shows  a  rise  from  100 
I  to  1  !•.!!,  the  arithmetic  backward  shows  a  fall  only 
from  l|f  to  94.46.  The  tw?  fffrm  a  bend  at  the  origin 
and  oneTor  both  ends  must  be  too  high.  This  tendency 
to  go  Higher  than  it  should  is  characteristic  of  the  arith- 
metic index  number. 

§  5.  The  Time  Reversal  Test  Expressed  Algebraically 

Algebraically,  .Test  1  (time  reversal)  may  be  stated  in 
general  terms  as  follows.  Let  the  two  dates  (or  two 
places)  be  distinguished  as  0  and  1  and  let  P0i  be  the/or- 
ward  index  number  of  prices,  i.e.  that  for  date  1,  relatively 
to  date  J)_  taken  as  a  base.  Then  Pi0  will  be  the  back- 
ward^ index  number,  i.e.  that  for  date  0  relatively  to  date  1 
taken  as  a  base.  With  this  notation  we  may  express  Test 
1  in  algebraic  terms  as  follows  :  P0i  X  PIO  should  =  1. 
This  is  the  same  as  saying  that  P0i  must  be  such  a  formula 
that  if  the  subscripts  0  and  1  be  interchanged,  the  new 
formula  resulting  will  become  the  reciprocal  of  the  old. 

The  failure  of  the  simple  arithmetic  index  number  to 
conform  to  Test  1  is  clearly  seen  if  we  examine  its  alge- 
braic expression.  If  we  take  the  year  designated  by  "0" 
as  the  base,  the  simple  arithmetic  index  number  for  year 
"l"is 

zfo) 

W 


n 

whereas,  if  we  reverse  the  comparison  by  taking  year  "1" 
as  the  base  (that  is,  interchange  the  subscripts)  the  sim- 
ple arithmetic  index  number  for  year  "0"  is 


2 


(S) 


n 
These  two  expressions  are  inconsistent  with  Test  1,  not 


72  THE  MAKING  OF  INDEX  NUMBERS 

being  reciprocals  of  each  other.    That  is,  they  are  not  of 
such  a  form  that  their  product  will  necessarily  be  unity. 

^ 


§  6.  The  Factor  Reversal  Test 

The  factor  reversal  test  is  analogous  to  the  time  reversal 
test.  Just  as  our  formula  should  permit  the  interchange 
of  the  two  times  without  giving  inconsistent  results,  so 
it  ought  to  permit  interchanging  the  prices  and  quan- 
tities without  giving  inconsistent  results  —  i.e.'  the  two 
Results  multiplied  together  should  give  the  true  value 
ratio. 

there  is  a  price  of  anything  exchanged/there 
a  quantity  of  it  exchanged,  or  produced,  or  con- 
sumed, or  otherwise  involved,  so  that  the  problem  of  an 
index  number  of  the  prices  implies  the  twin  problem  of  the 
index  number  of  the  quantities.  Thus  the  index  number 
of  the  prices  at  which  certain  commodities  are  sold  at 
wholesale  goes  hand  in  hand  with  the  index  number  of 
the  quantities  of  these  commodities  sold  at  wholesale. 
Likewise  we  find  paired  the  index  numbers  of  the  prices 
and  quantities  of  industrial  stocks  sold  on  the  New  York 
Stock  Exchange,  or  the  index  numbers  of  rates  of  wages 
and  of  the  quantities  of  labor  sold  at  those  rates  of  wages, 
«r  the  index  numbers  of  the  rates  of  discount  for  loans  and 
the  volume  «f  loans  made  at  those  rates  of  discount. 


§  7.  The  Simple  Arithmetic  Index  Number  Tested 
by  Factor  Reversal 

Of  the  28  formulae  thus  far  reached,  not  a  single  one 
conforms  to  Test  2  ! 

Numerically,  take  Formula  1,  the  simple  arithmetic, 
and  apply  it  to  an  example  which  is  simple  enough  to  fol- 
low through  in  detail.  Suppose  the  price  of  bacon  is 
twice  as  high  in  1918  as  in  1913  while  the  price  of  rubber 


TWO  GREAT  REVERSAL  TESTS  73 

is  exactly  the  same  in  1918  as  in  1913 ;  and  suppose  that 
the  quantity  of  bacon  sold  in  1918  is  half  as  much  as  the 
quantity  sold  in  1913  while  the  quantity  of  rubber  is  the 
same  in  both  years.  Evidently  the  value  of  bacon  sold  in 
1918  is  the  same  as  the  value  of  that  used  in  1913  (since 
half  the  quantity  of  bacon  is  sold  at  twice  TEe  price) 
and  likewise  the  value  of  the  rubber  remains  unchanged 
(since  both  its  price  and  quantity  remain  unchanged). 
Consequently,  the  total  value  of  both  together  remains 
unchanged  also,  A  good  index  number  of  these  prices 
multiplied  by  the  corresponding  index  number  of  these 
quantities  ought,  therefore,  to  give  (in  this  case)  100  per 
cent. 

With  these  figures  in  mind  let  us  test  the  mettle  of  the 
simple  arithmetical  average  by  applying,  it  alike  to  the 
above  prices  and  quantities.  By  this  formula  the  index 
number  of  prices  in  1918  as  compared  with  1913  is 

\200  +"100 

^  2       "  =  150  per  cent, 

and  the  index  number  of  quantities  is  \ 

-5o+ioo=75percent; 

""^•*  ^ 

Multiplied  together  these  results  give  112J  per  cent  in- 
stead of  the  true  100  per  cent.     Here  is  an  error  of  12J 
cent  either  in  the  index  number  of  prices,  or  in  that 
of  quantities,  or  shared  jointly  between  them. 
•  Again,  suppose  bread  doubles  in  price  and  triples  in 
quantity  so  that  its  value  sextuples,  and  butter  triples 
in  price  and  doubles  in  quantity  so  that  its  value  also 
sextuples ;  then  their  combined  value  certainly  sextuples. 
But  the  simple  arithmetic  index  number  would  make  it 

f)      I      O 

appear  that  bread  and  butter  had  increased  in  price    "      > 


74  THE  MAKING  OF  INDEX  NUMBERS 

O      I      C\ 

or  2  \  fold,  and  that  their  quantity  had  increased      J"    >  or 

« 

2|  fold,  according  to  which  their  values  are  represented 
to  have  increased  2£  X  2J,  or  6J  fold  instead  of  sixfold, 
the  true  figure. 

The  value  ratio,  unlike  an  index  number  of  prices  or  quan- 
tities, is  not  an  estimate  but  a  fact.  There  can  be  no  am- 
biguity about  it  or  any  question  of  reckoning  it  by  dif- 
ferent methods  as  in  the  case  of  index  numbers.  Thus, 
in  1913,  the  value  of  the  bacon  sold  was  its  price,  12.36 
cents  per  pound  multiplied  by  its  quantity,  1077  million 
pounds,  or  133  million  dollars.  In  the  same  way  the  value 
of  the  barley  sold  was  62.63  cents  per  bushel  X  178.2  mil- 
lion bushels  =  JJ2  million  dollars,  and  so  on  for  each  of  the 
other  34  commodities.  The  sum  total  of  these  36  prod- 
ucts, or  the  value  aggregate  in  1913  (Sgo^o)  is  13104.818 
million  dollars  and  can  be  nothing  else.  Likewise  for  the 

^f 

last  year,  1918,  the  value  aggregate  (Sp6g5)  is  29186.105 
and  can  be  nothing  else.  Thus  the  ratio  of  the  total  value 


of  1918  to  the  total  value  of  1913  is  ~          or  222.71  per 

lolOo. 

cent  and  can  be  nothing  else.     The  complete  table  of 
value  ratios  follows:  ^ 

TABLE  6.    VALUE  RATIOS  FOR  36  COMMODITIES 
1913-1918 


YEAR 

VALUE  RATIO 

1913  

100.00 

1914  

99.45 

1915  

108.98 

1916  

135.75 

1917  

192.23 

1918  

222.71 

These  are,  if  we  choose  to  call  them  so,  "  index  numbers  " 
of  the  total  or  aggregate  value.    But,  whereas  the  index 


TWO  GREAT  REVERSAL  TESTS  75 

numbers  of  prices  or  of  quantities  may  be  calculated  by 
many  different  methods,  the  comparative  merits  of  which 
are  debated  in  this  book,  the  " index  numbers"  of  value  are 
indubitable  and  undebatable.  They,  therefore,  afford  a 
fixed  rock  of  truth,  by  which  we  may  reckon  the  drifting 
courses  of  the  various  index  numbers  of  prices  and  quan- 
tities. The  problem  then  is  to  find  a  form  of  index 
number  such  that,  applied  alike  to  prices  and  quantities, 
it  shall  correctly  " factor"  any  such  value  ratio. 

Thus  we  can  say  with  absolute  certainty  that  the  total 
value  in  1918twas  223  per  cent  of  the  total  value  in  1913. 
But  when  we  ask  how  far  this  increase  from  100  to  223  rep- 
resents increased/prices  and  how   far  it  represents   in- 
\  creased   quantities,    we   enter    the    quagmire    of    index 
umbersx   We  are  searching  for  a  formula  which,  applied 
caprices,  will  really  measure  the  increase  of  the  prices, 
.d,  applied  to  quantities,   will  really  measure  the  in- 
rease  of  the  quantities ;  and  such  that  to  make  these 
wo  results  consistent,  their  product  should  give  the  re- 
'quired  223  per  cent* 

The  justification  for  Test  2  is  twofold :    (1)  no  reason 

Jean  be  given  for  employing  a  given  formula  for  one  of  the 

two  factors  which  does  not  apply  to  the  other,   and, 

(2)  such  reversibility  already  applies  to  each  pair  of  indi- 

.svidual  price  and  quantity  ratios,  and  should,  in  all  logic, 

apply  to  the  index  numbers  which  aim  to  represent  them 

in  the  mass. 

We  know  that  if  the  price  of  bread  in  1918  was  double 
its  price  in  1913  and  if  the  quantity  marketed  in  1918  was 
triple  that  in  1913  then  the  total  value  of  bread  marketed 
in  1918  was  six  times  that  marketed  in  1913.  By  anal- 
ogy we  have  a  right  to  expect  of  our  index  numbers,  if 
they  show  prices,  on  the  average,  to  have  doubled,  and 
quantities  to  have  tripled,  that  sixfold  correctly  repre- 
sents the  increase  in  total  value. 


76  THE  MAKING  OF  INDEX  NUMBERS 

Algebraically,  Test  2  is.> 


P(ty  formula  353)  X  Q(by  Formula  353) 
178 Z* 125%  =223* 


700% 


75 


77 


/ 


CHART  12.  The  product  of  the  price  index,  P,  times  the  quantity  index, 
Q,  both  calculated  by  the  same  formula  (No.  353)  equals  the  correct  "value 
\_ratio,"  V.  (In  this  ratio  chart,  therefore,  the  total  height  above  the 
origin,  100  per  cent,  of  the  point  in  the  chart  labeled  223  equals  the  sum 
of  the  heights  of  the  two  points  labeled  125  and  178,  above  the  same 
origin.) 


§  8.  The  Factor  Reversal  Test  Illustrated  Graphically 

Graphically,  Chart  12  shows  the  relation  of  index 
numbers  which  correctly  conform  to  Test  2.  It  shows 
how  one  of  the  index  numbers,  to  be  explained  later 
(No.  353),  fulfills  Test  2.  This  formula,  when  applied 
to  our  36  prices,  yields  178  per  cent  for  the  index  number 
in  1918,  relatively  to  1913  as  a  base,  and  when  applied 
to  quantities,  yields  125  per  cent;  and  -  these  two  figures 


TWO  GREAT  REVERSAL  TESTS 


77 


multiplied  together  give  correctly  the  true  value  ratio, 
223  per  cent,  as  given  in  Table  6 . 

Chart  13  shows  how  incorrect,  on  the  other  hand,  are 
the  index  numbers  calculated  by  Formula  9,  the  weighted 
arithmetic  average  in  which  the  weights  are  the  values 
in  the  given  or  current  year. 


P(by  formula  9)  X  Q (by  Formula  9)  not*  to  V   / 

not-to223%  / 


\ 

A 
/ 

/&  * 


223% 


I32K 


'&  '/4  '&  '&  77  78 

CHART  13.   Analogous  to  Chart  12  except  that  the  product  of  the  two 

iiuJex  numbers  is  not,  as  it  should  be,  equal  to  the  value  ratio.     The  dotted 

^Hne,  representing  the  product,  lies  above  the  true  value  by  a  percentage 

expressing  the  joint  error  of  the  two  indexes  (for  prices  and  for  quantities). 

§  9.  The  Factor  Reversal  Test  Reveals  a  Joint  Error 

Just  as  when  studying  Test  1,  we  checked  up  any  type 
of  index  number  by  noting  how  far  the  product  of  the 
index  number  reckoned  forward  by  the  index  number 


78  THE  MAKING  OF  INDEX  NUMBERS 

reckoned  backward  departed  from  unity,  so,  through 
Test  2,  we  check  up  by  noting  how  far  the  product  of 
the  price  ratio  (index  number  for  prices)  by  the  corre- 
sponding quantity  ratio  (index  number  for  quantities) 
departs  from  the  value  ratio. 

To  illustrate  how  great  this  error  may  be,  we  recur  to 
our  36  commodities.  We  know  that  the  total  value  of 
the  36  commodities  in  1917  was  $25,191,000,000  and  in 
1913  it  was  $13,105,000,000  so  that  the  true  value  ratio 
was  the  ratio  of  these  numbers,  or  192.23  per  cent.  But 
the  simple  arithmetic  index  number  (No.  1)  for  prices  for 
1917  relatively  to  1913  is  175.79  per  cent,  and  the  cor- 
responding index  number  for  the  same  dates  for  quan- 
tities is  125.84  per  cent.  The  product  of  these  two  is  221.21 
per  cent,  which  is  larger  than  the  truth  (192.23  per  cent) 
by  15.08  per  cent. 

This  is  an  exact  measure  of  the  inconsistency  of  the  two 
arithmetic  index  numbers  with  each  other  as  checked  up 
by  the  truth.  Thus  again  does  the  simple  arithmetic 
stultify  itself.  There  is  a  joint  error  here  of  15.08  per  cent 
somewhere,  just  as,  in  checking  up  by  Test  1,  we  found 
that  there  was  an  error  of  11.35  per  cent  somewhere. 
And,  just  as  before,  we  cannot  say  exactly  where  the  error 
lies.  The  15.08  per  cent  error  may  be  in  the  price  index, 
or  in  the  quantity  index,  or  it  may  be  shared  between  them. 

As  to  the  simple  geometric,  it  will  be  remembered  that 
we  could  not  convict  it  of  error  by  using  Test  1 ;  but,  by 
using  Test  2,  we  can  now  convict  it  of  error.  The  simple 
geometric  index  number  for  1917,  relatively  to  1913,  for 
prices,  is  166.65  per  cent  and,  for  quantities,  118.75  per 
cent ;  the  product  of  these  two  (instead  of  being  192.23 
as  it  should)  is  197.90,  which  is  2.95  per  cent  too  high. 

In  this  way,  by  means  of  Test  2,  we  can  convict  every 
pair  of  index  numbers  for  prices  and  quantities  in  our 


TWO  GREAT  REVERSAL  TESTS  79 

list,  as  thus  far  constituted,  of  some  degree  of  error.  Some 
formulae,  of  course,  come  much  nearer  than  others  to  con- 
forming to  Test  2.  The  least  joint  error  among  the  for- 
mulae thus  far  listed  is  53  's.  For  prices  for  1917  relatively 
to  1913  this  gives  162.07,  and  for  quantities,  119.36,  the 
product  of  which  is  193.45  per  cent  which  is  only  0.6  per 
cent  higher  than  the  required  192.23.  Incidentally  it 
may  be  noted  that  this  joint  error  of  53  P  and  53Q  is  the 
same  as  the  joint  error  we  found  by  Test  1  for  53P  and 
59P  and  is  the  same  as  the  joint  error  of  53Q  and  59Q. 

r  §  10.  The  Factor  Reversal  Test  Analogous  to 
the  Other  Reversal  Tests 

Algebraically,  the  various  sorts  of  reversibility  can  best 
be  seen  by  taking  some  particular  formula  as  an  example. 
Let  us  take  Formula  53  (Laspeyres').  For  prices  for- 
ward, Formula  53  is 


For  prices  backward  this  same  Formula  53  becomes 


the  "0"  and  "1"  being  reversed,  or  interchanged.    The 
two  above  applications  of  Formula  53  are  exactly  alike 
except  that  one  is  forward  in  time  and  the  other  is  back- 
ward.   Each  is  an  index  number  of  prices. 
Starting  again  with 


for  prices  forward,  let  us  this  time  interchange  or  reverse, 
not  the  "  0"  and  "1,"  but  the  "p's"  and  Vs."  We  then  get 


80  THE  MAKING  OF  INDEX  NUMBERS 

The  last  two  applications  of  Formula  53  are  exactly 
alike  except  that  one  is  for  prices  and  the  other  is  for  quan- 
tities. Each  is  a  forward  index  number. 

Thus  the  only  difference  between  the  two  tests  is 
that,  starting,  say,  with  the  Formula  53  for  prices  for- 
ward, 


for  Test  1  we  erase  "0"  wherever  it  occurs  and  write 
"1"  in  its  place,  and  vice  versa;  whereas,  for  Test  2,  we 
erase  "p"  wherever  it  occurs  and  write  "q"  in  its  place, 
and  vice  versa. 

Test  1  tells  us  that  after  the  specified  reversal  of  sym- 
bols, the  new  formula,  multiplied  by  the  old,  should 
give  unity,  i.e. 

v 


Test  2  tells  us  that  after  the  specified  reversal  of  sym- 
bols the  new  formula  multiplied  by  the  old  should  give 
the  value  ratio,  i.e. 


In  the  case  of  this  particular  formula,  (53)  neither  of 
these  equations  holds  true,  so  that  neither  test  is  fulfilled. 

While  we  are  noting  the  algebraic  interpretation  of 
Tests  1  and  2,  we  may  as  well  recur  to  the  "  Preliminary 
Test"  regarding  the  interchange  or  reversal  of  any  two 
commodities.  We  start  again  with 

Mo  +  p'lg'o  +  P"ig"o  +  •  •   • 


Pogo  +  p'og'o  +  p"og"o+  •  •  •' 
but  now  reverse  the  places,  not  of  "0"  and  "1"  nor  of 
"p"  and  '  V  but  of  "  '  "  and  "  "  "  (or  of  any  other 
two   accents  representing   two   different   commodities). 


TWO  GREAT  REVERSAL  TESTS  81 

That  is,  we  erase  "  '  "  wherever  it  occurs  and  write  "  "  " 
in  its  place,  and  vice  versa.    The  result  is  : 

+  p"ig"o  +  p'ig'o  +  •  •  • 


+  P'  Vo  +  pVo  +  •  •  • 

which  new  formula  is  (except  in  form)  the  same  as  the 
old  —  as  the  "  Preliminary  Test"  or  commodity  reversal 
test  requires.1 

Thus  the  commodity  reversal  test,  the  time  reversal  test, 
and  the  factor  reversal  test  alike  require  that  the  formula 
be  such  that  we  can,  with  impunity,  interchange  symbols. 
For  the  commodity  reversal  test  the  reversible  symbols  are 
the  commodity  symbols,  any  two  superscripts  such  as  "  '" 
and  """  ;  for  the  time  reversal  test  the  reversible  sym- 
bols are  the  two  time  symbols,  the  two  subscripts  "0"  and 
"1"  ;  and  for  the  factor  reversal  test  the  reversible  sym- 

1  bols  are  the  two  factor  symbols,  the  two  letters  "p"  and 
Reversibility  "with  impunity"    means    that    the 
results  of  such  reversal  shall  be  appropriate  to  the  case. 
For  commodity  reversal,  the  new  and  old  forms  of  the  for- 
mula ought  to  be  equal  ;  for  time  reversal,  they  ought  to 

'De^r_ej2ipr_p_cals  ;  for  factor  reversal,  they  ought  correctly 
to  "factor"  the^value,  ratio. 

These  three  tests  are  the  only  reversal  tests  possible, 
because  any  formula  for  an  index  number  contains  just 
three  sets  of  symbols,  the  letters,  the  subscripts,  the  super- 
scripts. The  three  reversal  tests  (Preliminary,  Test  1, 
and  Test  2)  merely  require  that  the  formula  shall  allow 
each  of  the  three  kinds  of  symbols  of  which  it  is  com- 
posed to  shift  about  with  impunity. 

As  these  requirements  of  reversibility  are  purely  formal 
and  mathematical,  they  evidently  have  a  very  wide  range 

1  This  test  is  met  by  all  the  formulae  in  this  book.  If  the  reader  wishes 
to  picture  a  case  where  this  test  would  not  be  fulfilled,  let  him  suppose  a 
minus  sign  in  place  of  one  of  the  plus  signs.  Also  see  §  1  above. 


82  THE  MAKING  OF  INDEX  NUMBERS 

of  application.  They  apply  to  any  index  number  — 
wholesale  prices,  retail  prices,  wages,  interest,  production, 
and  many  others  —  where  we  have  several  items  dis- 
tinguishable by  superscripts  such  as  "',"  "","  "'"," 
etc.,  two  times,  or  places,  or  other  groupings  distinguish- 
able by  two  subscripts,  such  as  "0"  and  "1,"  and  two 
magnitudes  distinguishable  by  two  letters  such  as  "p" 
and  "q"  after  the  analogy  of  the  case  we  just  took.1 

§  11.  Historical 

Test  1,  the  time  reversal  test,  seems  first  to  have  been 
used  by  Professor  N.  G.  Pierson  in  1896.2  Its  great  im- 
portance was  recognized  by  C.  M.  Walsh  in  190 1,3  and 
by  myself  in  191 1,4  as  well  as  by  other  writers. 

Unlike  Test  1,  Test  2  has  hitherto5  been  entirely  over- 
looked, presumably  because  index  numbers  of  quantities 
have  so  seldom  been  computed  and,  almost  never,  side 
by  side  with  the  index  number  of  the  prices  to  which  they 
relate.  Moreover,  the  analogy  between  the  three  kinds 
of  reversal  naturally  escaped  attention  since  most  users 
of  index  numbers  have  thought  in  concrete  terms  not 
algebraic ;  they  formed  a  mental  image  of  time  reversal 
only  from  the  calendar,  and  saw  no  advantage  in  pic- 
turing it  symbolically  as  an  interchange  of  "0"  and  "1" 
in  a  formula. 

1  See  Appendix  I  (Note  to  Chapter  IV,  §  10). 

2  Economic  Journal,  Vol.  vi,  March,  1896,  p.  128. 

8  Measurement  of  General  Exchange  Value,  pp.  324-32,  368-69,  389-90. 

*  Purchasing  Power  of  Money,  p.  401. 

6  It  was  first  formulated  in  the  paper,  of  which  this  book  is  an  expan- 
sion, read  December,  1920,  and  abstracted  in  "The  Best  Form  of  Index 
Number,"  Quarterly  Publication  of  the  American  Statistical  Association, 
March,  1921. 


CHAPTER  V 

ERRATIC,  BIASED,  AND  FREAKISH  INDEX  NUMBERS 

§  1.   Joint  Errors  between  Index  Numbers 

WE  have  seen  that  there  are  two  great  reversal  tests : 
(1)  that  the  product  of  forward  and  backward  indexes 
should  equal  unity,  and  (2)  that  the  product  of  price  and 
quantity  indexes  should  equal  the  value  ratio.  If  the 
former  product  is  not  equal  to  unity,  the  deviation  from 
unity  is  a  joint  error  of  the  forward  and  backward  in- 
dexes ;  and,  likewise,  if  the  latter  product  is  not  equal  to  the 
value  ratio,  the  deviation  from  that  figure  is  a  joint  error 
of  the  price  and  quantity  indexes. 

Tables  7  and  8  —  one  for  each  test  —  show  the  joint 
errors  of  each  of  the  28  formulse.  Take,  for  instance, 
the  index  numbers  for  prices  and  quantities  as  between 
1913  and  1917.  Under  Test  1,  the  error  lies  jointly  be- 
tween the  index  for  1917  relatively  to  1913,  and  the  index 
for  1913  relatively  to  1917,  when  both  are  reckoned  by  any 
given  formula ;  while  under  Test  2  the  error  lies  jointly 
between  the  price  index  and  the  quantity  index,  when  both 
are  reckoned  by  any  given  formula. 

It  will  be  seen  that  the  joint  errors  vary  from  zero  to 
nearly  30  per  cent  (for  Formula  11, 1918,  Test  2) ;  and  that 
Formulae  7,  9,  13,  15  show  very  large  joint  errors,  while 
those  of  3,  5,  17,  19,  53,  59  are  among  the  smallest.  Not 
a  single  one  of  the  28  formulse  is  entirely  free  from  one 
or  the  other  of  the  joint  errors,  and  only  four  (21,  31,  41, 
51)  are  free  from  either  error.  These  four  conform  to 
Test  1.  (Each  of  the  weighted  modes,  43,  45,  47,  49, 

83 


84 


THE  MAKING  OF  INDEX  NUMBERS 


has  too  small  a  joint  error  under  Test  1  to  be  measured 
by  the  rough  method  used  for  calculating  them.)  In 
other  words,  every  one  of  these  formulae  is  certainly 
erratic,  as  revealed  by  the  two  tests.  It  may,  of  course, 

TABLE  7.  JOINT  ERRORS  OF  THE  FORWARD  AND  BACK- 
WARD APPLICATIONS  OF  EACH  FORMULA  (THAT  IS, 
UNDER  TEST  1)  IN  PER  CENTS 

(PRICE  INDEXES) 

Example :  The  first  figure,  +  1.19,  is  found  as  follows :  The  index  num- 
ber forward  X  the  index  number  backward  (both  by  Formula  1)  =  96.32 
per  cent  X  105.06  per  cent  =  101.19  per  cent  as  compared  with  the  truth, 
100  per  cent  —  an  error  of  +1.19  per  cent. 


FORMULA 
No. 

1914 

(PER  CENTS) 

1915 

(PER  CENTS) 

1916 

(PER  CENTS) 

1917 

(PER  CENTS) 

1918 

(PER  CENTS) 

1 

+  1.19 

+2.56 

+3.83 

+  11.34 

+  8.68 

3 

-0.39 

-0.43 

-0.24 

+  0.63 

+  0.25 

5 

+0.39 

+0.43 

+0.24 

-  0.63 

-  0.25 

7 

+0.90 

+3.73 

+6.08 

+24.53 

+  12.07 

9 

+  1.68 

+4.59 

+6.56 

+22.78 

+  11.03 

11 

-1.17 

-2.50 

-3.69 

-10.19 

-  7.99 

13 

-1.65 

-4.39 

-6.15 

-18.55 

-  9.93 

15 

-0.90 

-3.60 

-5.73 

-19.70 

-10.77 

17 

-0.39 

-0.43 

-0.24 

+  0.63 

+  0.25 

19 

+0.39 

+0.43 

+0.24 

-  0.63 

-  0.25 

21 

0. 

0. 

0. 

0. 

0. 

23 

-1.01 

-2.42 

-4.14 

-  9.60 

-  4.99 

25 

-0.26 

-1.59 

-2.80 

-10.75 

-  5.53 

27 

+0.26 

+  1.62 

+2.88 

+12.05 

+  5.85 

29 

+  1.02 

+2.48 

+4.32 

+  10.62 

+  5.26 

31 

0. 

0. 

0. 

0. 

0. 

33 

-0.41 

-0.58 

-1.75 

-  4.71 

-  5.04 

35 

-0.13 

-0.24 

-1.29 

-  2.23 

-10.15 

37 

+0.13 

+0.24 

+  1.30 

+  2.29 

+  11.30 

39 

+0.41 

+0.58 

+1.78 

+  4.95 

+  5.31 

41 

0. 

0. 

0. 

0. 

0. 

43 

o.± 

O.=t 

0.=*= 

o.± 

0.* 

45 

o.± 

o.± 

o.± 

o.± 

0.=*= 

47 

o.± 

O.=t 

O.=b 

o.± 

0.  =fc 

49 

0.  =*= 

o.± 

O.=t 

o.± 

o.± 

51 

0. 

0. 

0. 

0. 

0. 

53 

-0.39 

-0.43 

-0.24 

+  0.63 

+  0.25 

59 

+0.39 

+0.43 

+0.24 

-  0.63 

-  0.25 

BIASED  INDEX  NUMBERS 


85 


TABLE  8.  JOINT  ERRORS  OF  THE  PRICE  AND  QUANTITY 
APPLICATIONS  OF  EACH  FORMULA  (THAT  IS,  UNDER 
TEST  2)  IN  PER  CENTS 

(FORWARD  INDEXES) 

Example:  The  first  figure,  —3.85,  is  found  as  follows :  The  index  num- 
ber for  price  X  the  index  number  for  quantity  (both  by  Formula  1)  =96.32 
per  cent  X  99.27  per  cent  =  95.617  per  cent,  as  compared  with  the  true 
value  ratio,  99.45  per  cent  —  an  error  of  —3.85  per  cent  of  the  true  99.45. 


FORMULA 
No. 

1914 

(PER  CENTS) 

1915 

(PER  CENTS) 

1916 

(PER  CENTS) 

1917 

(PER  CENTS) 

1918 

(PER  CENTS) 

1 

-3.85 

+2.19 

+  12.73 

+  15.08 

+  5.40 

3 

-0.39 

-0.43 

-  0.24 

+   0.63 

+  0.25 

5 

+0.39 

+0.43 

+  0.24 

-  0.63 

-  0.25 

7 

+  1.55 

+4.53 

+  5.67 

+  18.44 

+  11.92 

9 

+2.26 

+5.53 

+  6.47 

+16.62 

+  10.58 

11 

-8.01 

-5.66 

+  3.27 

-  8.27 

-29.50 

13 

-2.51 

-4.46 

-  4.96 

-12.58 

-11.18 

15 

-1.67 

-3.80 

-  4.81 

-14.02 

-11.90 

17 

-0.39 

-0.43 

-  0.24 

+  0.63 

+  0.25 

19 

+0.39 

+0.43 

+  0.24 

-  0.63 

-  0.25 

21 

-5.84 

-1.86 

+  7.79 

+  2.95 

-  7.22 

23 

-1.40 

-2.57 

-  3.62 

-  6.53 

-  5.22 

25 

-0.61 

-1.79 

-  2.46 

-  7.81 

-  5.61 

27 

+0.60 

+  1.87 

+  2.51 

+  8.74 

+  5.91 

29 

+  1.35 

+2.81 

+  3.19 

+  7.40 

+  5.08 

31 

-0.66 

-3.55 

+  2.41 

+  1.02 

+  4.02 

33 

-0.85 

-5.04 

-  8.85 

-  5.69 

-  7.05 

35 

-0.49 

-4.42 

-  8.72 

-  3.21 

-  7.15 

37 

+0.04 

-2.37 

-  6.83 

+  3.80 

+  4.74 

39 

+0.23 

-1.78 

-  6.65 

+  2.46 

-  1.23 

41 

-5.77 

-9.26 

-13.60 

-19.16 

+  3.83 

43 

-1.94 

-5.66 

-18.18 

-16.33 

-  6.67 

45 

-1.94 

-5.66 

-18.18 

-16.33 

-  6.67 

47 

-1.94 

-5.66 

-18.18 

-16.33 

-  6.67 

49 

-1.94 

-5.66 

-18.18 

-16.33 

-  6.67 

51 

-1.28 

-0.92 

-  5.98 

-  7.41 

+  4.61 

53 

-0.39 

-0.43 

-  0.24 

+  0.63 

+  0.25 

59 

+0.39 

+0.43 

+  0.24 

-  0.63 

-  0.25 

be  erratic  beyond  these  revelations,  as  a  small  joint  error 
may  be  the  net  effect  of  large  but  offsetting  errors  in  the 
two  index  numbers  for  which  that  joint  error  is  revealed. 


86  THE  MAKING  OF  INDEX  NUMBERS 

We  shall  find  reasons  for  believing  this  to  be  true  of  the 
modes  particularly. 

§  2.  Bias,  under  Test  1,  Inherent  in  Arithmetic  and 
Harmonic  Types  of  Formulae 

But,  in  many  cases,  we  can  convict  a  formula  not  only 
of  being  erratic  when  tested  by  Test  1,  but  also,  under 
that  test,  of  being  distinctly  biased,  i.e.  subject  to  a  fore- 
seeable tendency  to  err  in  one  particular  direction.  Under 
Test  1,  four  formulae  conform  (21,  31,  41,  51) ;  six  (which 
reduce  to  two  when  duplicates  are  excluded)  are  merely 
erratic  (3,  5, 17, 19,  53,  59) ;  and  18  are  biased.  Of  these 
18,  the  following  nine  have  an  upward  bias :  1,7,  9,  27, 
29,  37,  39,  47,  49,  while  the  following  nine  have  a  down- 
ward bias :  11,  13,  15,  23,  25,  33,  35,  43,  45. 

All  cases  of  provable  bias  are  under  Test  1.  Let  us 
begin  with  Formula  1.  It  can  be  proved  that  the  prod- 
uct of  this  formula,  applied  forward  and  backward, 
instead  of  being  unity,  as  required  by  Test  1,  always  nec- 
essarily exceeds  unity. 

Numerically,  that  this  is  true  in  any  given  case,  can 
readily  be^seen  by  trial.  Thus,  suppose  two  commodities 
of  which  the  forward  price  ratios  are  100  and  200  per 
cent,  and  the  backward,  therefore,  100  per  cent  and  50 
per  cent.  We  are  to  show  that 

(100  +  200)      (lt)0  +  50) 

9 X~ 9~~ 

2  2 

exceeds  unity.    This  is  150  per  cent  X  75  per  cent,  or  113 
per  cent,  which-  exceeds  unity  by  13  per  cent. 

Algebraically,  the  proof  that  the~  "product^  of  the  arith- 
metic forward  by  the  arithmetic  backward  Always  and 
necessarily  exceeds  unity  is  given  in  the  Appendix.1 
1  See  Appendix  I  (Note  to  Chapter  V,  §  2). 


BIASED  INDEX  NUMBERS  87 

Thus,  Formula  1,  the  simple  arithmetic  average,  has  nec- 
essarily a  positive  joint  error.  While  we  cannot  go  further 
and  say,  in  any  given  case,  how  much  of  this  error  lies 
in  its  forward  form  and  how  much  in  its  backward  form, 
in  the  absence  of  any  reason  to  accuse  the  one  more  than 
the  other,  we  are  jusjifigj.  in  accusing  both  equally.  The 
proportionate  share  of  the  total  necessary  error  thus  pre- 
sumed to  belong  to  each  is  called  its  "bias."  In  Table  7, 
the  bias  of  the  index  number  of  prices,  by  Formula  1,  for 
the  36  commodities,  is,  for  1917,  one-half  of  11.34  per 
cent,  or  about  5|  per  cent.1  That  is,  the  arithmetic 
average  exhibits  an  inherent  tendency  to  exaggeration, 
a  "bias,"  such  that,  in  the  instance  cited,  it  yields  a  re- 
sult probably  too  high  by  about  5 \  per  cent. 

This  inherent  tendency  in  the  arithmetic  type  always 
exists  irrespective  of  the  method  of  weighting  used. 
So  long  as  the  same  weights  are  used  forward  and  back- 
ward, the  product  of  the  arithmetic  forward  and  backward 
will  exceed  unity.  The  reasoning  in  the  Appendix,  above 
cited,2  applies  to  the  arithmetic  index  number  as  such, 
whether  simple  or  weighted.  By  similar  reasoning,  it 
may  be  sho^wn  that  the  harmonic  index  number,  with  or 
without  any  given  weighting,  has  an  inherent  bias  down- 
ward. iJiatHs,  its  forward  and  backward  forms,  multi- 
plied t^getlpi',  give  a  result  always  and  necessarily  less 
than  unity. '\  The  joint  error  is  the  difference  between 
unity  andHfite  product  of  the  harmonic  forward  by  the 
harmonic  backward.  ,  V>«>  ^ 

Graphically ',ithe  intimate  relationship  between  the  arith- 
metic and  harmonic  bias  (which  are,  at  bottom^  the  same) 

1  The  mathematical  reader  will  prefer  to  reckon  the  equal  shares  more 
precisely,  i.e.  in  equal  proportions  instead  of  equal  parts  (i.e.  Vl.1134  -  1).  -•'• 
But  the  result  is,  of  course,  approximately  the  same. 

2  Appendix  I  (Note  to  Chapter  V,  §  2).  x 


88  THE  MAKING  OF  INDEX  NUMBERS 

is  clearly  seen  in  Charts  14P  and  14Q  made  from  Chart  11. 
By  reversing  the  direction  of  the  dotted  line  representing 

the  simple  arith- 

Type  Bias  of  Formula  Na  I        metic  backward, 
(Prices)  we  rePresent  its 

reciprocal.  But 
this  reciprocal 
turns  out  to  be 
the  simple  har- 
monic forward. 

simple  arithmetic  ^^^~~~$imple  harmonic    Thus     the     chart 
backward    ^--^  forward 

^^-— "  T         shows     that     the 

failure,  previously 
y?  78  pointed  out,  of  the 

CHART  14P.  The  simple  harmonic  forward  the  arithmetics  for- 
same  reversed,  as  the  simple  arithmetic  back-  ward  and  back- 
ward. The  bias  is  half  the  gap  at  the  right. 


the  prolongation  of  the  other  is  precisely  the  same  thing 
as  the  failure  of  the  arithmetic  forward  and  the  harmonic 
forward  to  coincide  with  each  other.  The  use  of  the  har- 
monic enables  us  to  get  rid  of  all  backward  lines  and  merely 
contrast  forward  lines.  Consequently,  the  joint  error  of 
the  arithmetic  (i.e.  the  deviation  from  unity  of  the  product 
of  the  forward  and  backward  arithmetics),  previously  pic- 
tured as  the  bend  between  two  lines  which  ought  to  be  pro- 
longations of  each  other,  is  now  pictured  as  the  angle  be- 
tween two  forward  lines  which  ought  to  coincide.  Half  of 
this  divergence  represents  the  upward  bias  of  the  arith- 
metic and  half  the  downward  bias  of  the  harmonic. 

§  3.   Joint  Error  Expressible  by  Product  or  Quotient 

Thus,  the  joint  error,  either  of  the  arithmetic  or  of  the 
harmonic,  may  be  written  in  two  ways.  The  old  way 
was  as  the  difference  between  unity  and  the  product  of  the 


BIASED  INDEX  NUMBERS  89 

arithmetic  forward  by  the  arithmetic  backward.  The  new 
way  is  as  the  difference  between  unity  and  the  quotient 
of  the  arithmetic  ^  _.  /  A/  / 

forward  by  the  TypeBias  or  Formula  Nal 

harmonic  forward  (Quantities) 

which,  as  stated, 

is  the  reciprocal    t 

of  the  arithmetic    ''7  s;mpie  arlfhni^c 

backward.  These       --Backward   slmplc  ari1hmetic 

two     alternative  "^^^^    forward      ^ 

ways  of  exhibit-  ^ — ^^simple  harmonic 

ing  the  joint  er-    \5% 

ror  are  important  //  'J8 

enough  to  be  for-  CHART  14Q.     Analogous  to  Chart  14P. 

mulated    mathe- 
matically.    That  is,  the  old  way  is 

arithmetic  forward  X  arithmetic  backward  exceeds  unity. 
But  we  may  substitute  for  " arithmetic  backward"  its 
equal,  "the  reciprocal  of  the  harmonic  forward,"  giving 

arithmetic  forward  X  harmonic  forward  exceeds  unity, 

or,  more  briefly, 

arithmetic  forward  ^^  ^ . 
harmonic  forward 

Similarly  the  joint  error  of  the  harmonic  may  be  written 
either  as 

harmonic  forward  X  harmonic  backward  is  less  than  unity 

or,  as  follows,          ^rmonic  forward  ^  ^  ^  ^ . 
arithmetic  forward 

The  new,  or  quotient,  form  is  in  each  case  the  more  con- 
venient and  obviates  the  need  of  using  any  backward 
index  numbers. 


90  THE  MAKING  OF  INDEX  NUMBERS 

But,  while  the  quotient  form  is  the  easier  to  handle  and 
much  the  more  convenient  to  use  in  computations  and 
charts,  the  product  form  affords  the  more  convincing 
proof  of  bias.  If  only  the  quotient  form  were  mentioned, 
it  might  be  hastily  inferred  that  our  only  reason  for  ascrib- 
ing an  upward  bias  to  the  arithmetic  and  a  downward 
bias  to  the  harmonic  is  that  the  former  exceeds  the  latter. 
But  the  argument  goes  much  deeper.  The  argument  is 
not  merely  that  one  of  two  index  numbers  exceeds  an- 
other. The  point  is  that  the  harmonic  essentially  repre- 
sents an  arithmetic  backward.  .  We  ascribe  an  upward 
bias  to  the  arithmetic  solely  on  the  showing  of  the  arith- 
metic itself  —  because  the  arithmetic  forward  multiplied  by 
the  arithmetic  backward  is  always  greater  than  unity. 
In  this  product  form  the  reasoning  does  not  require  the 
introduction  of  the  harmonic,  or  any  other  type  of  aver- 
age than  the  arithmetic.  Even  if  we  had  never  heard  of 
any  other  average  than  the  arithmetic,  it  would  stand 
convicted  on  its  own  testimony.  The  same  argument, 
of  course,  applies  to  the  harmonic,  without  invoking  the 
arithmetic  average.  In  short,  the  harmonic  is,  as  it  were, 
a  concealed  arithmetic,  and  so  either  may  be  made  to  dis- 
appear and  give  place  to  the  other. 

Graphic  Resume  of  Type  Bias 

Graphically,  Charts  15P  and  15Q  show  three  principal 
types  of  index  numbers  compared.  There  are  five  groups, 
each  from  a  separate  origin:  one  group  (at  the  top) 
representing  the  simple  index  numbers  and  four  groups 
representing  the  index  numbers  having  the  four  weight- 
ings respectively.  Each  group  contains  all  three  types, 
so  that  there  are  15  formulae  in  all.  We  observe  that,  in 
each  group,  the  geometric  always  lies  about  midway  be- 
tween the  arithmetic  and  the  harmonic,  and  that  this 


BIASED  INDEX  NUMBERS  91 

is  true  of  the  index  numbers  in  the  chain  systems  (shown 
by  the  balls)  as  truly  as  in  the  fixed  base  systems  (shown 
by  the  curves  themselves).  The  upward  bias  of  the  arith- 
metic and  the  downward  bias  of  the  harmonic  manifest 
themselves  in  every  case.  In  each  group  the  three  curves 
have  the  same  weighting:  1,  11,  21,  —  simple;  3,  13, 
23,  —  weighting  7  ;  5, 15,  25,  —weighting  II ;  7, 17,  27,  — 
weighting  777;  9,  19,  29,  —  weighting  IV.  The  three 
differ  only  in  type,  and,  in  each  case,  the  arithmetic  type 
is  the  highest  and  the  harmonic,  the  lowest.  The  wide 
gap  between  the  arithmetic  and  harmonic  in  each  case 
represents  their  joint  error  (by  the  quotient  method),  and 
so  measures  the  upward  bias  of  the  arithmetic  and  down- 
ward bias  of  the  harmonic. 

§  5.  Bias  in  the  Weighting 

The  kind  of  bias  just  described  inheres  in  the  arith- 
metic and  harmonic  types  of  average.  But  there  is  another 
kind  of  bias  inhering  in  the  system  of  weighting  used  and 
affecting  all  the  weighted  formulae  thus  far  described, 
except  the  aggregatives.  That  is,  weight  bias  applies  to 
any  type  of  index  number  susceptible  of  value  weighting. 
The  weights  of  the  aggregative  are,  of  course,  not  values 
but  mere  quantities,  as  has  been  explained. 

To  illustrate  weight  bias,  take,  for  example,  the  geo- 
metric index  number.  We  know  that  the  geometric  type, 
as  such,  has  no  bias,  and  it  will  be  remembered  that  the 
simple  geometric  obeys  Test  1  (being  merely  erratic  under 
Test  2).  But  when  we  weight  the  geometric  under,  for 
instance,  system  7F,  we,  at  once,  impart  an  upward  bias. 
Empirically,  this  is  proved  by  the  fact  that,  if  we  take 
this  geometric  7F,  both  forward  and  backward,  the  prod- 
uct is  invariably  found  to  exceed  unity. 

Again,  as  we  have  seen,  the  arithmetic  type,  as  such, 


92 


THE  MAKING  OF  INDEX  NUMBERS 


Three  Types  of  Index  Numbers 
of  Prices 


i/ 

21 

^tf 


Arithmetic 

Geometric 

Harmonic 


/        ///  ^5 


77 


78 


CHART  15P.  The  geometric  always  lies  about  midway  between  the 
arithmetic  and  harmonic,  whether  fixed  base  or  chain.  The  five  groups  are 
separated  to  save  confusion,  really  forming  five  distinct  diagrams.  The 
gap  of  each  arithmetic  and  harmonic  from  the  middle  is  its  type  bias. 

does  have  a  bias.    But  when  we  weight  the  arithmetic 
under  system  IV  we  impart  an  additional  bias ;  its  bias 


BIASED  INDEX  NUMBERS 

Three  Types  of  Index  Numbers 
of  Quantities 


93 


Arithmetic 

Geometric 

Harmonic 


&  7*  &  70  77 

CHART  15Q.    Analogous  to  Chart  15P. 


78 


is  approximately  doubled  thereby.  Empirically,  this  is 
proved  by  the  fact  that,  if  we  take  this  arithmetic  7F, 
both  forward  and  backward,  the  product  is  invariably 
found  to  exceed  unity  by  about  twice  the  bias  of  the 


94  THE  MAKING  OF  INDEX  NUMBERS 

simple  arithmetic.  In  this  way,  by  actual  trial,  we  can 
convince  ourselves  of  the  truth  of  the  proposition  that  the 
weighting  systems  /  or  II  impart  a  downward  bias  to 
any  index  number,  while  777  and  IV  impart  an  upward 
bias. 

§  6.   Outline 1  of  Argument  as  to  Geometric, 
Median,  and  Mode 

Besides  such  empirical  evidence,  good  logical  reasons 
for  this  weight  bias  exist ;  but  they  are  not  so  simple  to 
set  forth  as  were  the  reasons  for  the  arithmetic  and  har- 
monic type  bias,  chiefly  because  the  weight  bias,  with 
which  we  now  have  to  deal,  unlike  type  bias,  with  which 
we  dealt  in  previous  sections,  is  partly  a  matter  of  mere 
probability.  In  studying  weight  bias  it  will  be  more  con- 
venient to  take  up  the  quotient  method  first.  We  shall 
see: 

(1)  For    any    given    type    of    formula    having    value 
weights,  the  index  numbers  with  weightings  /  or  II  are, 
in  general,  smaller  than  the  index  numbers  with  weight- 
ings 777  or  IV ; 

(2)  These   inequalities   are   partly   necessary,    partly 
probable.     That  the  index  number  weighted  7  is  less  than 
777  and  that  77  is  less  than  IV  are  mathematically  neces- 
sary.    But  that  7  is  less  than  IV  and  77  than  777  can- 
not be  proved  to  be  absolutely  necessary  but  only  to  be 
highly  probable ; 

(3)  Since,  then,  IV  exceeds  7,  and  777  exceeds  77, 
the  quotient  of  IV  divided  by  7  exceeds  unity,  as  does  777 
divided  by  77.     These   excesses  may  provisionally   be 
called  joint  errors.     Such  a  joint  error  allotted  in  equal 
proportions  to  index  numbers  weighted  7  and  7F,  or  to 

1  For  details  of  the  argument  in  this  and  the  following  section  see  Ap- 
pendix I  (Note  to  Chapter  V,  §  6),  which  may  best  be  read  after  reading 
the  text. 


BIASED  INDEX  NUMBERS  95 

index  numbers  weighted  II  and  III,  gives  each  index 
number  its  bias; 

(4)  Weight  bias  is  most  simply  seen  in  the  case  of  the 
weighted  geometries,  medians,  and  modes  because  these 
have  weight  bias  only,  uncomplicated  by  type  bias.  In 
these  cases  the  quotient  form  of  weight  bias  is  easily  de- 
rived from  the  product  form,  and  vice  versa.  Let  us  take, 
for  instance,  the  geometries  I  and  IV,  or  Formulae  23 
and  29,  and  express  the  weight  bias  of  29.  The  quotient 
form  of  this  bias  is  half  the  excess  above  unity  of  the 
quotient  ff  (both  indexes  being  forward  or  both  back- 
ward). This  excess  will  be  found  to  be  identical  with 
half  the  excess  above  unity  of  the  product  of  29  forward 
X  29  backward.  Likewise,  if  we  take  25  and  27,  the 
weight  bias  of  27  is  half  the  excess  above  unity  of  f£,  which 
is  the  same  as  half  the  excess  above  unity  of  27  forward 
X  27  backward. 

As  previously  stated,  the  product  form  is  the  preferable 
one  to  use  in  our  logic  because  it  employs  only  one  for- 
mula. Thus  it  makes  29  convict  itself  of  error  by  con- 
fronting it,  as  it  were,  by  its  own  reversed  image  in  the 
looking  glass. 

i  •  The  foregoing  relate  to  the  four  systems  of  weighting 
as  applied  to  the  geometries,  medians,  or  modes.  The 
weight  biases  of  the  geometric  compare  closely  in  magni- 
tude with  those  found  for  the  type  bias  of  the  arithmetic 
and  harmonic. 

§  7.   Supplementary  Argument  as  to  Arithmetic 
and  Harmonic 

With  the  weighted  arithmetic  and  weighted  harmonic,  the  case  is  more 
complex.  Take  arithmetic  IV,  Formula  9,  or  Palgrave's  formula.  The 
(forward)  arithmetic  IV,  divided  by  (forward)  arithmetic  /,  Formula  3, 
is  here  not  identical  with  the  arithmetic  IV  forward  multiplied  by 
arithmetic  IV  backward,  because  type  bias  complicates  the  situation. 


96  THE  MAKING  OF  INDEX  NUMBERS 

The  product  mentioned  (arithmetic  IV  forward  by  arithmetic  IV  back- 
ward) is  identical  with  the  quotient  of  arithmetic  IV  forward  divided  by 
the  harmonic  7,  Formula  13  forward.  That  is,  A  IV  for.  X  A  IV  back. 

A  IV  A  IV 

(or  9  for.  X  9  back.)  is  not  identical  with  (or  f)  but  with 

A  7  H  7 

(or  -fs).  But  we  know,  from  our  study  of  type  bias,  that  the  harmonic 
7  lies  below  arithmetic  /  and,  in  fact,  that  their  joint  error  is  the  excess 
above  unity  of  arithmetic  /  divided  by  harmonic  7.  Hence  we  find,  from 
our  present  study  of  weights,  that  arithmetic  IV  exceeds  arithmetic  7; 
and,  from  our  former  study  of  types,  that  arithmetic  7  exceeds  harmonic 
7.  It  follows  that  arithmetic  IV  doubly  exceeds  harmonic  7.  Conse- 
quently, it  is  doubly  true  that  arithmetic  IV  divided  by  harmonic  7  ex- 
ceeds unity.  But  this  is  the  same  thing  as  saying  that  it  is  doubly  true 
that  arithmetic  IV  forward  multiplied  by  arithmetic  IV  backward  ex- 
ceeds unity. 

Thus,  we  convict  arithmetic  IV  by  itself,  although  as  a  step  in  our 
reasoning  we  included  the  type  joint  error  of  arithmetic  7  and  harmonic  7. 
That  is,  of  the  decreasing  series  :  arithmetic  IV,  arithmetic  7,  harmonic  7, 
the  first  exceeds  the  third  by  a  joint  error,  not  only  in  the  quotient  sense 
but  also  in  the  product  sense;  likewise,  the  second  exceeds  the  third  by 
a  joint  error,  not  only  in  the  quotient  sense  but  also  in  the  product  sense; 
but  the  first  exceeds  the  second,  by  a  joint  error,  only  in  the  quotient 
sense.  That  is,  the  total  excess  of  arithmetic  IV  over  harmonic  7  is 
type  and  weight  bias,  in  the  product  sense ;  part  of  this  total  excess, 
namely,  that  of  arithmetic  7  over  harmonic  7,  is  type  bias  in  the  product 
sense;  hence,  indirectly,  the  remaining  excess,  i.e.  that  of  arithmetic  IV 
over  arithmetic  7,  is  weight  bias  in  the  product  sense.  Thus,  the  arith- 
metic IV  has  a  double  dose  of  upward  bias,  part  of  its  bias  being  due  to 
its  being  of  the  arithmetic  type  and  part  being  due  to  its  having  weighting 
IV.  The  same  is  true  of  arithmetic  777;  while  the  harmonics  7  and  77 
have  a  double  dose  of  downward  bias. 

§  8.  The  Argument,  Numerically,  Algebraically, 
and  Graphically 

I  have  outlined  these  steps  of  reasoning,  partly  to 
help  the  reader  who  chooses  to  follow  the  argument  in 
the  Appendix l  in  detail,  and  partly  to  make  it  unneces- 
sary to  do  so  for  readers  who  do  not  so  choose.  Here,  for 
brevity,  I  will  merely  indicate  the  results  by  actual  figures. 

Numerically,  then,  we  can  see  how  the  matter  works 
out  by  repeating  here  the  weights  of  selected  commodi- 
ties under  the  four  systems  of  weighting. 

1  See  Appendix  I  (Note  to  Chapter  V,  §  6). 


BIASED  INDEX  NUMBERS 


97 


TABLE  9.     THE  FOUR  SYSTEMS  OF  WEIGHTING  THE 

PRICE  RELATIVES  FOR   1917,  ^,    ^T,   ETC. 

Po     p'o 


COMMODITY 

WEIGHTING 
SYSTEM  I 
poqo 

WEIGHTING 
SYSTEM  // 
Pofli 

WEIGHTING 
SYSTEM  /// 
MH 

WEIGHTING 
SYSTEM  IV 
PtQt 

/////,   also 
IV/II,  also 

P_4 
PO 

(in  millions  of  dollars) 

(in  per  cents) 

Bituminous  coal  .  . 
Coke               

606 
140 
462 
422 

701 
172 
577 
596 

1708 
494 
1203 
715 

1976 
604 
1502 
1011 

282 
352 
260 
170 

Pig  iron  

Oats 

Anthracite  coal  .  .  . 
Petroleum 

35 
1282 
98 
1971 

40 
1835 
147 
1916 

39 
1292 
80 
2290 

44 
1848 
123 

2227 

111 

101 
83 
116 

Coffee    
Lumber         

Thus,  bituminous  coal  rose  in  price  from  100  to  282,  and 
the  weights  under  systems  I  and  777  are  606  and  1708, 
which  are  also  exactly  in  the  ratio  of  100  to  282  (or,  again, 
the  weights  under  77  and  IV  are  701  and  1976  —  also  ex- 
actly as  100  to  282).  Thus,  the  last  column  not  only 
gives,  in  each  case,  the  price  relative,  or  price  rise,  but  also 
the  weight  rise  (i.e.  the  ratio  of  777  to  7  and  of  IV  to  77). 
Algebraically,  the  reason  for  this  last-named  result  is 
clear.  As  the  headings  of  the  columns  indicate,  the 
weights  under  777  and  7  are  ptq0,  etc.,  and  p0q0,  etc.,  and 


the  ratio  of  these  weights,         .^  reduces,  by  cancellation, 


to  ^,  which  is  identical  with  the  price  relative.    Thus 
Po 

the  greater  the  price  relative,  the  more  heavily  is  it 
weighted  under  system  777  (as  compared  with  system  7) 
and  in  exact  proportion.  Under  system  777  (as  compared 
with  system  7)  the  rule  is  "to  him  that  hath  shall  be 


THE  MAKING  OF  INDEX  NUMBERS 


given"  —  that  is,  the  high  price  relatives  draw  relatively 
high  weights  and  the  low,  low.     Consequently,  the  high 

Four  Methods  of  Weighting  Compared 

By  base  year  vetues  (p>ty.  etc) 

mixed          ••  (fty     ~ ) 

»  •»  •»  (p.fy    " ) 

••  given  year       ••  (p.y.     •») 

(Prices) 


CHART  16P.  When  the  price  elements  in  the  weights  are  changed,  the 
index  number  is  greatly  changed,  and  in  a  foreseeable  direction.  When 
the  quantity  elements  are  changed,  the  index  number  is  scarcely  altered, 
and  in  no  foreseeable  direction.  The  weight  bias  is  hah7  of  a  gap. 
(Changing  the  price  elements  is  as  between  curves  3  and  7,  13  and  17,  23 
and  27,  or  between  5  and  9,  15  and  19,  25  and  29.  Changing  the  quantity 
elements  is  as  between  3  and  5,  13  and  15,  23  and  25,  or  between  7  and  9,  17 
and  19, 27  and  29.  There  are  three  distinct  diagrams.  Hereafter,  the  reader 
will  be  expected  to  distinguish  for  himself  between  separate  diagrams  on 
the  same  chart,  by  the  fact  that  they  have  separate  origins.) 

price  relatives  have  more  influence  on  the  resulting  index 
number  (which  is  an  average  of  all  the  price  relatives) 
than  under  system  7,  and,  therefore,  make  the  resulting 


BIASED  INDEX  NUMBERS 


99 


index  number  larger  than  that  resulting  under  system  /. 
Likewise,  IV  has  exactly  the  same  contrast  with  II. 

It  is  clear,  then,  that  under  systems  III  and  IV  the 
high  price  relatives  are  heavily  weighted  and  so  dominate 
their  average  (the  index  number),  i.e.  raise  it;  or,  if  we 
prefer  to  say  so,  under  systems  /  and  //  the  low  price 

Four  Methods  of  Weighting  Compared 
(Quantities) 


\5. 


73  74  75  '/<?  77  73 

CHABT  16Q.    Analogous  to  Chart  16P  (interchanging  "  price  "  and 
"quantity"). 

relatives  are  heavily  weighted  and  so  dominate  their 
average,  i.e.  lower  it.  The  cards  in  the  weighting  are 
stacked  so  that  weighting  I  or  II  pulls  the  index  number 
down,  or  weighting  III  or  IV  pushes  it  up,  or  both.  The 
former  weighting  has  a  bearish,  as  the  latter  has  a  bullish, 
influence,  or  both,  and  in  the  absence  of  any  other  data 
and  with  no  reason  to  believe  the  error  all  one  way,  we 
can  best  describe  the  tendency  as  a  "bias"  in  the  weight- 


100        THE  MAKING  OF  INDEX  NUMBERS 

ing ;  an  upward  bias  for  II I  and  IV  and  a  downward  for 
I  and  II. 

Thus,  in  an  index  number  of  prices  the  price  element 
in  the  weight  has  far  greater  influence  on  the  result  than 
the  quantity  element.  We  need  not  trouble  much  as  to 
the  quantity  element,  but  we  must  take  great  pains  to 
see  that  the  price  element  is  what  it  should  be.  Instead 

Four  Methods  of  Weighting  Compared 
(Prices) 


15* 


'&  '/+  '/5  76  77  78 

CHABT  17P.  The  effects  on  the  index  number  of  changing  the  weight- 
ing are,  in  the  case  of  the  median,  similar  to,  but  smaller  and  more  erratic 
than,  the  effects  in  the  cases  of  the  arithmetic,  harmonic,  and  geometric. 
In  some  years  the  agreement  is  closer  than  is  the  case  in  the  arithmetic, 
harmonic,  and  geometric,  but  when  there  is  a  difference  it  is  apt  to  be 
much  more  pronounced. 

of  having  to  "mind  our  p's  and  <?'s"  we  need  only  mind 
our  "p's" !  But  for  the  quantity  indexes  the  opposite 
holds. 

Graphically,  weight  bias  manifests  itself  in  Charts  16P 
and  16Q  in  each  of  the  three  groups  of  curves.  In  each 
group  the  four  curves  are  of  the  same  type  and  differ 


BIASED  INDEX  NUMBERS 


101 


only  in  weighting.  It  will  be  noted  that  the  curves,  end- 
ing in  3  or  5  (weightings  I  and  77),  always  practically 
coincide,  as  do  the  curves  ending  in  7  or  9  (weightings 
777  and  77),  although  there  is,  in  all  three  cases,  a  wide 
gap  between  the  former  pair  on  the  one  hand  and  the 
latter  pair  on  the  other.  The  mystery  of  this  persist- 
ently recurring  gap  representing  the  joint  error  (by  the 
quotient  method)  is  to  be  solved  by  the  existence  of  a 

Four  Methods  or  Weighting  Compared 
(Quantities) 


\5% 


'/+  75  Jtf  77 

CHART  17Q.     Analogous  to  Chart  17P. 


73 


distinct  upward  bias  of  777  and  IV  and  downward  bias  of 
7  and  77. 

Charts  17 P  and  17 Q  (upper  diagrams)  show  the  medi- 
ans, which  exhibit  the  same  sort  of  biases,  though  less  than 
in  Chart  16,  and  resemble  the  two  medians  of  Chart  10. 
But  we  notice  a  curious  and  important  difference  between 
these  charts  and  Charts  16P  and  16Q  for  the  arithmetic, 
harmonic,  and  geometric.  In  all  these  preceding  cases 
the  curves  ending  in  7  and  9,  for  example,  nearly,  but 
not  quite,  coincided  with  each  other,  according  as  slight 
changes  in  the  incidence  of  weighting  produced  corre- 


102         THE  MAKING  OF  INDEX  NUMBERS 

spondingly  slight  effects.  But,  in  the  case  of  the  median, 
the  effects  of  changed  weighting  go  by  fits  and  starts.  In 
most  instances  curves  37  and  39,  for  instance,  stick  even 
closer  together  than  7  and  9,  or  17  and  19,  or  27  and  29. 
But  when  the  cleavage  between  them  is  broken  at  all  they 
are  apt  to  be  torn  wide  apart.  This  characteristic  of  the 
median,  its  insensitiveness,  as  contrasted  with  the  arith- 
metic, harmonic,  and  geometric,  has  already  been  referred 
to.  The  four  modes  (not  charted)  are  indistinguishable. 

§  9.  Double  Bias  Illustrated  Numerically  and 
Graphically 

Double  Blds(W&ghtBias and          Numerically,  our 

Type  Bias)  of  Formula  Na9       ^st  illustration  of 

*r  .  double  bias  is  per- 

(Prices)  haps  that  of  Pal- 

grave's  index  num- 
i   ker  (Formula  9  in 
sfflzr;    our     series),      the 
arithmetic  weighted 


IV,    weighted    by 
I5*      given  year  values, 

,_  m  Pi<li>      etc.       This 

index  has    a    very 

CHART  18P.     Showing,  by  the  divergence  .    . 

between  9  forward  and   9    backward,   their  large      joint      error 

joint  error,  half  of  which  is  the  upward  bias  under  Test  1  which 
of  Formula  9.     This  divergence,  or  joint  error, 

is  also  shown  by  the  divergence  between  curves  we  are  to    analyze. 

9  and  13.     In  this  form  it  is  easily  subdi-  Jn  Table  7  we  find 

vided  into  three  parts  of  which  the  middle  is  ,  -,        .    .    ,  /. 

negligible.     Of  the  rest,  half  is  upward  bias  the   joint   error   tor 

of  9,  comprising  two  parts,  weight  bias  and  this    Palgrave     for- 
type  bias  (the  weight  bias  being  half  of  the  ,  pnnlipH   fn 

divergence  between  9  and  5,  and  the  type  bias  1      **»  d 

being  half  of  the  divergence  between  3  and  1917    relatively     to 

13).     The  other  two  quarters  of  the  whole    i  QI  o     4.,.    u~    oo  7Q 
...         ,1        .     .,        ?    ,    j  j    j      11       IcJlO.     to     DC     £4*t<y 

constitute  the  similar,  but  downward,  double 

bias  of  13.  per  cent.      For 


BIASED  INDEX  NUMBERS  103 

1918  relatively  to  1917,  it  is  6.99  per  cent.  That  is,  Pal- 
grave's  index  number  taken  forward  multiplied  by  Pal- 

grave'sindexnum-  A    .,  0.    /i,/-.^o.       mti 

Eer   taken    back-  Double  Bw(We,ght  fas  1*4 

ward  is  1  +  .0699.  TypeBias)  of  Formula  A/a  9 

About  half  of  this  (Quantities) 

error  of  6.99  per 

cent,   or  3.5   per 

cent,  may  be  as- 

signed  to  each  of 

the  two  forms  (for-    ***£*  73 

ward    and    back- 

j\       ITT       I.    11  CHAET  18Q.    Analogous  to  Chart  18P. 

ward).     We  shall 

find  that  this  3.5  per  cent  error  is  in  turn  made  up  of 
three  parts. 

Graphically,  Charts  18P  and  18Q  show  the  whole  joint 
error  and  the  three  parts  into  which  it  may  be  divided. 
The  lines  are  numbered  with  the  identification  numbers 
and  also  lettered  ("A"  for  arithmetic,  "H"  for  har- 
monic, with  the  Roman  numerals  attached  to  indicate 
the  system  of  weighting).  Beginning  with  Palgrave's 
formula  (9,  or  arithmetic  IV)  taken  forward,  let  us  also 
take  it  backward,  as  shown  by  the  dotted  line,  likewise 
labeled  "9"  or  "A  77."  These  two  applications  of  Pal- 
grave's  formula,  forward  and  backward,  multiplied  to- 
gether do  not  give  unity.  In  other  words,  the  forward 
and  backward  lines  are  not  prolongations  of  each  other. 
The  prolongation  forward  of  the  backward  line  gives 
us  13  or  H  I,1  and  the  divergence  between  9  and  13  (i.e.  the 
vertical  distance  between  the  right-hand  ends  of  lines  9 
and  13  in  the  chart)  represents  the  percentage  joint  error 
of  9  forward  and  backward,  that  is,  .0699. 

This  joint  error  consists  of  three  parts.  Practically 
1  For  proof  see  Appendix  I  (Note  to  Chapter  V,  §  9). 


104         THE  MAKING  OF  INDEX  NUMBERS 

the  upper  half,  that  is,  the  divergence  between  9  and  5, 
is  due  to  changing  the  weighting  of  the  arithmetic  from 
IV  (as  used  in  9)  to  II  (as  used  in  5),  i.e.  from  p\q\, 
etc.,  to  po^i,  etc.,  i.e.  by  changing  the  price  element  in 
the  weighting.  The  next  part  is  very  small  and  due  to 
changing  further  the  weighting  system  from  II  (as  used 
in  5)  to  7  (as  used  in  3),  i.e.  from  pQqi,  etc.,  to  p0q0, 
etc.,  i.e.  by  changing  the  quantity  element  in  the  weights. 
Finally,  the  third  part,  practically  the  lower  half,  is  due 
to  changing  from  the  arithmetic  type  (A  7  or  3)  to  the 
harmonic  type  (H  7  or  13),  while  retaining  the  same 
weighting  system  (7). 

Recapitulating,  we  note  three  shifts :  (1)  a  shift  of  the 
price  element  in  the  weights,  (2)  a  shift  of  the  quantity 
element  in  the  weights,  and  (3)  a  shift  of  the  type  of  aver- 
age. The  middle  shift  is  always  almost  negligible  and 

may  be  either  up 

weight  Bias  of 'formula  Na 29       Or  down.    Both 
(Prices)  the  other   shifts 

are       necessarily 
down  in  the  or- 


we 

them.     The  first 
shift     represents 
I555        a   joint  error  of 
arithmetic  IV 
J8  and  77  (9  and  5), 

Chart  19  P.     Showing  by  the  divergence  be-  half      being  the 

tween  29  forward  and  29  backward  their  joint                  ,     ,  .  - 
error,  half  of  which  is  the  upward  bias  of  For-  upward 

mula  29.     This  divergence  or  joint  error  is  also    weighting  IV 

shown  by  the  divergence  between  curves  29  and          *       Vmlf  +V»o 

23.     In  this  form  it  is  easily  subdivided  into  two    £                 clu>  ~ 

parts,  of  which  the  lower  is  negligible,  and  half    downward  bias 

of   the   upper   is    the    upward  bias  of  29  with    ^   jj       The  last 
weighting  IV,  and  the  other  half  the  downward 

bias  of  23  with  weighting  7.  shift  represents  a 


BIASED   INDEX  NUMBERS  105 

joint  error  of  the  arithmetic  and  harmonic  types,  half  be- 
ing the  upward  bias  of  the  arithmetic,  and  half,  the  down- 

ward  bias  of  the       weight  Bias  of  Formula  No.29 

harmonic. 

By  a  different  (Mhe*> 

choice  of  lines  the 
analysis  may  be 
presented    some- 
what differently,    29(sa)- — " 
but  the  essential  e/7 

fact    Will    always  CHABT  19Q'     Analogous  to  Chart  19P. 

appear  that  9,  or  A  IV,  has  a  double  dose  of  up- 
ward bias,  first,  because  it  is  of  the  arithmetic  type  and, 
secondly,  because  its  system  of  weighting  is  IV,  while 
13,  or  H  7,  has  a  double  dose  of  downward  bias,  being 
both  harmonic  and  weighted  by  system  7. 

The  example  chosen  illustrates  both  kinds  of  bias, 
weight  bias  and  type  bias.  Only  the  arithmetic  and  har- 
monic formulae  have  type  bias,  consequently  the  corre- 
sponding diagrams  for  the  geometric,  median,  and  mode 
are  simpler,  as  there  is  no  type  bias.  Charts  19P  and 
19Q  show  the  contrast  between  weightings  7  (23)  and  IV 
(29)  for  the  geometric. 

We  see,  then,  that  the  joint  errors  shown  in  Tables 
7  and  8  are  not  altogether  unaccountable  or,  as  we  may 
say,  accidental ;  but  are,  in  two  instances,  due  to  clearly 
discernible  causes.  First,  the  arithmetic  and  harmonic 
index  numbers  have  a  definite  bias,  upward  and  downward 
respectively,  and,  secondly,  the  methods  of  weighting  777, 
IV,  on  the  one  hand,  and  7,  77,  on  the  other,  have  like- 
wise an  upward  and  downward  bias  respectively.1 

1  The  reader  should  not  forget  that  all  these  results  are  general ;  they 
hold  good  whether  prices  are  rising  or  falling ;  they  are  not  due  to  any  se- 
lection of  commodities  (other  than  that  self-selection  by  which,  e.g.  under 
given  year  weighting,  high  price  relatives  draw  high  weights). 


106         THE  MAKING  OF  INDEX  NUMBERS 


§  10.  The  Five-tined  Fork 

Charts  20P  and  20Q  (upper)  give  a  bird's-eye  view  of 
how  the  four  methods  of  weighting  affect  the  three  princi- 
pal types  of  index  numbers,  arithmetic,  harmonic,  and  geo- 


Five -Tine  Fork 
of  IB  Curves 
(Prices) 


•18 


CHART  20P.  The  five-tined  fork  given  in  Chart  9P,  with  additional 
curves,  and  their  factor  antitheses  (lower  dotted  diagram),  which  arrange 
themselves  in  the  inverse  order  of  the  originals.  The  four  gaps  are  biases. 

metric,  exhibiting  both  single  and  double  bias.  We  can 
see  substantially  the  same  five-tined  fork  as  in  Charts  9P 
and  9Q,  where  only  weights  I  and  IV  were  used.  But  in 
Charts  20P  and  20Q  No.  5  is  added  to  No.  3,  and  almost 
coincides  with  it,  7  almost  coincides  with  9,  15  with  13, 


BIASED   INDEX  NUMBERS 


107 


17  with  19,  25  with  23,  27  with  29 ;  also  3  and  5  coincide 
absolutely  with  17  and  19  respectively.1 

The  middle  tine  is  the  bottom  of  the  arithmetic  index 
numbers  (weighting  7,  or  curve  3,  and  77,  or  curve  5), 
and,  at  the  same  time,  it  is  the  top  of  the  harmonic 

The  Fivc-Tme  Fork 
of  18  Curves 
(Quantifies) 


t 


7J  74  75  '/$  17 

CHART  20Q.  Analogous  to  Chart  20P.  The  spacing  of  the  upper  dia- 
gram is  equal  and  opposite  to  that  of  the  lower  part  of  20P;  that  of  the 
lower  is  equal  and  opposite  to  that  of  the  upper  part  of  20P. 

(weighting  777,  or  17,  and  IV,  or  19),  while  the  other  two 
arithmetics  (weighting  777,  or  7,  and  IV,  or  9)  are  at 
the  extreme  top,  and  the  other  two  harmonics  (weighting 
7,  or  13,  and  77,  or  15)  are  at  the  extreme  bottom.  The 
extreme  upper  and  lower  tines  represent  doubly  biased 
index  numbers.  The  geometries,  as  in  Charts  9P  and 
9Q,  having  single  bias,  lie  astride  of  the  central  tine. 

1  The  reader,  at  this  point,  may  disregard  the  curves  numbered  103  and 
upward,  and  also  all  the  even-numbered  curves.  These  will  be  referred 
to  later. 


108         THE  MAKING  OF  INDEX  NUMBERS 

That  is,  the  geometries  with  weightings  777  and  IV  (27  and 
29)  lie  substantially  midway  within  the  arithmetic  two- 
tined  fork,  while  those  with  weightings  7  and  77  (23  and 
25),  likewise  midway  within  the  harmonic  two-tined  fork. 

§  11.    Bias  Depends  on  Dispersion 

All  the  various  formulae  for  any  year  would,  of  course, 
agree  in  their  results  if  all  the  price  relatives  for  that  year 
happened  to  agree.  The  more  nearly  the  price  relatives 
coincide,  the  more  nearly  the  averages  will  coincide,  and 
the  more  the  price  relatives  scatter  or  disperse,  the  more 
the  formulae  can  be  expected  to  disagree.  It  is  interest- 
ing, therefore,  to  trace  the  effect  which  the  dispersion 
of  the  original  data  has  on  the  disagreement  between 
index  formulae,  and,  especially,  on  the  disagreement  be- 
tween the  biased  formulae. 

The  relation  between  bias  and  dispersion  is  not  a  re- 
lation of  simple  proportion.  Thus,  in  the  period  1914- 
1917,  the  bias  increased  quite  out  of  proportion  to  the 
dispersion.  Nevertheless  a  definite  formula  can  be  given 
connecting  any  bias  with  the  dispersion  of  the  price  rela- 
tives.1 When  the  dispersion  is  small,  the  bias  is  very 
small  indeed.  This  explains  why  the  bias  of  the  arith- 
metic type  has  not  been  clearly  discerned  by  users  of  index 
numbers.  As  shown  in  the  table  given  in  the  Appendix, 
the  average  dispersion  of  the  price  relatives  above  and  be- 
low their  mean  must  reach  about  20  per  cent  to  make  the 
bias  as  much  as  1.67  per  cent.  But  if  the  prices  disperse 
30  per  cent,  or  half  as  much  again,  the  bias  doubles.  And 
when  the  average  dispersion  is  50  per  cent  the  bias  reaches 
8.34  per  cent.  When  the  dispersion  reaches  100  per 
cent,  i.e.  when  the  high  price  relatives  are,  on  the  average, 

1  See  Appendix  I  (Note  to  Chapter  V,  §  11),  where  methods  of  measur- 
ing dispersion  are  given,  with  formulae  and  tables. 


BIASED  INDEX  NUMBERS 


109 


double  their  mean  and  the  low  price  relatives  are,  on  the 
average,  half  of  their  mean,  the  bias  reaches  25  per  cent. 
(Any  "mean  "  will  do.)  Thus,  if  we  know  the  dispersion, 
we  can  tell  how  biased  an  arithmetic  index  number  may 
be  in  any  given  case  and  approximately  correct  it. 

§  12.   Our  36  Commodities  Disperse  Unusually  Widely 

It  is  in  time  of  war,  crises,  or  other  disturbance  that 
the  dispersion  of  prices  is  likely  to  be  great.  Consequently, 
the  arithmetic  index  number  is  the  most  untrustworthy 
for  such  periods,  e.g.  through  1861-1875  and  1914-1922. 
It  is  chiefly  from  the  last-named  period  that  our  data  for 
the  prices  and  quantities  of  the  36  commodities  were 
taken.  They  disperse  very  widely,  therefore,  as  com- 
pared with  the  dispersion  we  find  in  any  peace  time 
period  of  the  same  length.  Table  10  shows  the  average 
dispersion  of  our  36  price  relatives  and  of  36  of  Sauer- 
beck's price  relatives  (the  36  commodities  most  nearly 
comparable  to  our  36) : 

TABLE   10.    DISPERSION1  OF  36  PRICE    RELATIVES, 
(1)   BEFORE  THE  WORLD  WAR,  AND  (2)  DURING  IT 

(In  per  cents) 


YEAB 

SATJERBECK'S 

YEAR 

THIS  BOOK'S 

1846  

base 

1913  

base 

1856  

20 

1914  

10 

1866  

44 

1915  . 

16 

1876  

29 

1916  

24 

1886  

25 

1917 

58 

1896  

28 

1918  

45 

1906  

35 

1913 

42 

1  Measured  by  the  (arithmetically)  calculated  "standard  deviation,"  as 
explained  in  Appendix  I  (Note  to  Chapter  V,  §  11). 


110 


THE  MAKING  OF  INDEX  NUMBERS 


It  will  be  noted :  (1)  that  in  the  four  years,  1913  to 
1917,  the  dispersion  reached  58  per  cent,  which  was  more 
than  any  figure  reached  in  the  entire  period  of  67  years, 
1846  to  1913 ;  (2)  that,  in  both  series,  war  increases  dis- 
persion, the  Civil  War  year,  1866,  having  the  highest 
figure  in  the  first  column ;  (3)  that  the  return  of  peace  re- 
duces the  dispersion,  as  witness  the  figures  for  1876  and 
1918 ;  and  (4)  that,  in  general,  there  is  a  progressive  in- 
crease in  dispersion  with  the  lapse  of  time,  as  witness  the 
figures  for  1886  to  1913.  These  same  points  are  always 
in  evidence  whatever  period  is  examined. 

The  preceding  table  relates  to  prices  only.  Unfortu- 
nately there  are  no  quantity  relatives  associated  with 
Sauerbeck's  price  figures.  But  Professors  Day  and  Per- 
sons have  worked  out  quantity  figures  for  12  crops. 
Table  11  shows  the  dispersion  of  the  quantities  of  the  36 
commodities  and  of  the  12  crops  studied  by  Professors 
Day  and  Persons : l 

TABLE  11.    DISPERSION2  OF  36  AND  OF  12, 
QUANTITY  RELATIVES 

(In  per  cents) 


YEAR 

36  COMMODITIES 

YEAE 

12  CHOPS 

1913  

base 

1880  

38 

1914  

12 

1885  

25 

1915  

17 

1890 

26 

1916  

17 

1895  

25 

1917  

24 

1900  

18 

1918 

27 

1905 

16 

1910     ... 

base 

1915  

18 

1920  

10 

1  See  Edmund  E.  Day,  "An  Index  of  the  Physical  Volume  of  Produc- 
tion," The  Review  of  Economic  Statistics,  pp.  246-59,  September,  1920. 

2  Measured  by  the    (geometrically)   calculated  "standard   deviation," 
weighted  as  explained  in  Appendix  I  (Note  to  Chapter  V,  §  11). 


BIASED  INDEX  NUMBERS  111 

It  will  be  seen  that  the  dispersion  of  the  36  quantities 
reaches,  in  only  five  years,  a  figure  higher  than  that 
reached  in  a  span  of  25  years  for  the  12  crops.  The  only 
instance  in  which  the  12  crop  dispersion  reaches  a  higher 
figure  is  in  1880,  30  years  away  from  the  base. 

It  is  because  of  the  unusually  great  dispersion  of  our 
36  prices  and  quantities  that  these  data  afford  a  very 
severe  test  of  accuracy  of  the  conclusions  reached  in  this 
book.  The  index  numbers  which  we  have  calculated 
and  shall  calculate,  whether  biased,  freakish,  or  merely 
slightly  erratic,  differ  among  themselves  much  more  than 
they  would  during  six  years  of  peace.  Thus  in  Table  7 
the  biased  Formula  1  has  a  joint  error  of  11.34  per  cent, 
calculated  forward  and  backward  between  1913  and  1917, 
only  four  years  apart.  But  Professor  A.  W.  Flux  x  shows 
that  Sauerbeck's  index  number  calculated  forward  and 
backward  between  two  periods,  ten  years  apart  (one  of  the 
two  being  the  period  1904-1913  and  the  other  the  year 
1919),  gives  a  discrepancy  of  only  eight  per  cent ;  he  also 
shows  that  the  Board  of  Trade  index  number  calculated 
forward  and  backward  between  1871  and  1900,  a  span  of 
29  years,  gives  a  discrepancy  of  13  per  cent,  which  is  only 
a  little  more  than  the  11.34  per  cent  we  find  here,  although 
covering  seven  times  as  long  a  period  of  time. 

§  13.   Formulae  may  be  Erratic  without  being  Biased 

In  the  case  of  Palgrave's  index  number  (Formula  9 
above  discussed)  the  two  kinds  of  bias  —  type  bias  and 
weight  bias  —  conspire,  as  we  have  seen,  to  raise  the 
index  number  and  the  same  is  true  of  Formula  7.  Like- 
wise for  Formulae  13  and  15  the  two  conspire  downward. 

For  Formulae  3  and  5  ( the  same  as  Formulae  17  and  19), 
on  the  other  hand,  the  two  types  of  bias  almost  exactly 

1  Journal  Royal  Statistical  Society,  March,  1921,  p.  174. 


112         THE  MAKING  OF  INDEX  NUMBERS 

offset  each  other.  Thus,  Formula  3,  by  virtue  of  being 
arithmetic,  has  an  upward  bias,  but,  by  virtue  of  having 
weighting  /,  has  also  a  downward  bias ;  likewise  as  to 
Formula  5.  As  there  is  no  way  of  telling  which  of 
the  two  opposing  tendencies  will  be  the  greater,  the  net  re- 
sult may  be  said  to  be  unbiased,  though  still  erratic, 
for  bias  is  a  foreseeable  tendency  to  err  in  one  direction. 

Again,  taking  the  same  formulae  considered  as  harmon- 
ics, we  may  say  that  Formula  19,  being  harmonic,  is 
biased  downward,  but,  being  weighted  by  system  IV  is 
also  biased  upward ;  and  likewise  as  to  Formula  17.  Or, 
taking  the  same  formulae  considered  as  aggregatives  (for 
3  is  the  same  as  53,  and  19  as  59),  we  may  say  that  For- 
mula 53  has  no  bias ;  for,  while  it  is  one-sided  in  that  it 
contains  the  quantities  for  only  one  of  the  two  years, 
the  other  being  omitted,  we  cannot  ordinarily  foretell 
whether  this  fact  will  raise  or  lower  the  index  number ; 
and  likewise  as  to  Formula  59.  We  can,  however,  say 
that  the  Formulae  53  and  59  are  slightly  erratic;  for, 
taken  forward  and  backward,  the  product  is  not  unity 
though  very  close  to  it,  as  Tables  7  and  8  show.  Thus 
the  weighted  aggregatives,  or  their  equivalent  arithmetics 
and  harmonics,  are  erratic  without  being  biased;  some 
other  random  selection  of  commodities  than  those  here 
chosen  might  show  a  negative  error  in  place  of  a  posi- 
tive error  in  our  tables,  and  vice  versa.  Thus,  we  must 
distinguish  sharply  between  index  numbers  like  Formula 
51,  which  are  simply  very  erratic,  and  those  like  Formula 
9  or  Formula  13,  which  are  very  much  biased. 

§  14.  Erratic  and  Freakish  Index  Numbers 

It  may  be  assumed,  for  the  present  at  least,  that  all  index 
numbers  are  erratic  to  some  degree.  One  of  the  chief 
objects  of  this  book  is  to  show  to  what  degree. 


BIASED  INDEX  NUMBERS  113 

Tables  7  and  8  convict  every  one  of  the  28  index  num- 
bers so  far  considered  of  some  error.  In  the  case  of  18 
of  the  formulae  we  can  show  reason  for  some  at  least  of 
the  errors,  the  part  which  has  been  described  and  dis- 
cussed as  "bias."  In  the  cases  of  the  other  ten  formulae, 
the  joint  errors  shown  are  "accidental"  in  the  sense  that 
we  can  assign  no  reason  beforehand  for  their  being  in 
one  direction  or  the  other.  Thus  as  to  Formula  7,  which 
shows  in  Table  7  a  joint  error  in  1917  of  +24.53  per  cent 
under  Test  1  (and  of  +18.44  per  cent  under  Test  2  in 
Table  8)  and  a  positive  joint  error  in  the  ten  columns 
of  the  two  tables,  we  can  confidently  predict 1  that  we 
shall  always  find  a  positive  joint  error  whatever  the  data 
may  be  which  enter  the  formula.  But,  as  to  Formula 
21  which,  under  Test  2,  shows  a  joint  error  in  1916  of 
+7.79  per  cent  and  in  1918  of  -7.22  per  cent,  there 
can  be  no  assurance  whether,  for  any  other  particular 
set  of  data,  the  joint  error  will  be  positive  or  negative. 
All  we  can  say  is  that  21  is  certainly  erratic. 

Nor  can  we  infer  from  these  tables  what  the  whole 
error  of  any  formula,  whether  biased  or  merely  erratic, 
really  is.  Thus  from  Table  7  we  find  that  Formula  43, 
the  mode,  with  base  year  weighting,  shows  an  imper- 
ceptible joint  error,  and  21,  31,  41,  51,  no  error  at  all. 
But  this  may  be  due  to  the  fact  that  errors  forward  and 
backward  happen  to  offset  each  other.  That  this  is  the 
case  is  proven  by  Table  8  which  finds  errors  in  all  these 
formulae,  that  for  43  reaching  —18.18  per  cent  in  1916. 
Thus,  if  the  real  error  under  Formula  43,  in  the  price 
index  forward,  in  1916,  is  —5.44  per  cent  and  backward 
the  same,  while  the  real  error  in  the  quantity  index  for- 
ward is  —13.34  per  cent  and  backward  the  same,  the 

1  From  the  analysis  in  Appendix  I  (Note  to  Chapter  V,  §  2)  and 
Appendix  I  (Note  to  Chapter  V,  §  6). 


114         THE  MAKING  OF  INDEX  NUMBERS 

figures  in  both  tables  would  be  explained.  As  a  matter 
of  fact  these  errors  are  the  real  errors  of  No.  43  —  in- 
dubitable within  a  small  fraction  of  one  per  cent.  But 
we  are  not  yet  ready  to  show  this. 

Thus  a  small  joint  error,  being  only  a  net  error  between 
two  index  numbers,  is  compatible  with  large  errors  in 

/nsensi five/less  of  Median  and  Mode 
to  Number  of  Commodities 


\5% 


35     7     9     II     13    15    17 .  19    21    23   25  27   29   31    33  3$ 

NUMBER   OF  COMMODITIES 

CHART  21.  Showing  that  a  change  in  the  number  of  commodities  from 
3  commodities  to  5,  7,  etc.,  commodities  seldom  affects  the  median  (31)  and 
mode,  even  the  weighted  mode  (43  and  47) .  Both  median  and  mode  remain  the 
same  throughout  the  sixteen  changes  in  the  number  of  commodities,  ex- 
cept for  six  changes  in  the  median  (two  at  the  *)  and  two  in  the  mode. 
Wlien  the  mode  does  change,  it  changes  violently. 

both.    But  how  can  we  ascribe  individual  error  otherwise 
than  by  dividing  by  two  the  joint  errors  in  the  tables  ? 

While  the  answer  to  this  question  must  be  given  in 
stages  or  instalments,  we  can,  at  this  point,  show  that 
the  modes,  which  never  have  perceptible  bias,  are  never- 
theless very  erratic,  and  the  medians,  which  seldom  have 
much  bias,  are  moderately  erratic.  The  evidence  lies  in 
the  fact  that  the  mode  and,  to  a  less  extent,  the  median, 


BIASED  INDEX  NUMBERS  115 

are  insensitive  to  many  of  the  factors  of  which  an  index 
number  is  expected  to  be  a  sensitive  barometer. 

The  introduction  of  a  new  commodity  ought,  evidently, 
to  change,  in  some  degree,  any  price  index  which  pretends 
to  be  a  sensitive  expression  of  the  data  from  which  it  is 
computed  (unless,  of  course,  the  new  commodity  happens 
to  have  a  price  relative  exactly  equal  to  the  index 
number).  But  the  mode  and  median  often  remain  un- 
changed, like  the  hands  of  a  clock  not  rigidly  connected 
with  the  wheels  which  are  supposed  to  move  it. 

Again,  every  change  in  weights  will  be  reflected  by  a 
change  in  any  truly  sensitive  index  number.  But  the 
mode  will  often,  in  fact  usually,  remain  inert  even  when 
the  weighting  is  changed  radically. 

Graphically,  both  these  points  are  illustrated  by  Chart 
21  which  traces  median  and  mode  numbers  through  suc- 
cessive stages  as,  one  after  another,  we  introduce  new 
commodities,  beginning  with  three  (lime,  pig  iron,  and 
eggs)  taken  by  lot  and  adding  successively  new  commodi- 
ties by  lot,  two  by  two,  until  all  36  are  introduced.  The 
median  taken  is  the  simple  median ;  the  mode  is  weighted. 
The  weighted  median  is  not  taken  because  it  is  somewhat 
sensitive  to  weighting  and  we  are  illustrating  insensitive- 
ness. 

We  see  that  in  all  the  17  stages  at  which  there  ought  to 
be  a  change  the  median  changes  only  six  times  and  the 
mode  only  twice!  No  clock  can  keep  time  to  the  second 
if  it  jumps  only  once  in  a  minute,  or  once  in  an  hour. 
Such  a  clock  must  invariably  be  in  error  most  of  the  time, 
although,  from  the  clock  itself,  we  cannot  say  how  much. 
In  short,  the  horizontal  lines  in  the  diagram  betray  the 
existence  of  error,  but  not  how  much  error.  Further- 
more, as  to  the  mode,  the  fact  that  Formulae  43,  45,  47, 
49  can  all  be  represented  by  the  same  curve  shows  that 


116         THE  MAKING  OF  INDEX  NUMBERS 

the  mode  pays  no  attention  to  big  changes  in  weighting, 
thus  further  betraying  error.  When  an  index  number 
is  highly  erratic  we  have  called  it  freakish.  Evidently  the 
modes,  even  the  weighted  modes,  are  freakish  and  the 
median,  likewise,  though  in  less  degree. 

Formula  51  is  freakish  for  another  reason.  Instead 
of  being  insensitive  to  influences  which  ought  to  affect 
it,  it  is  sensitive  to  influences  which  ought  not  to  affect 
it.  Evidently  an  index  number,  to  be  a  true  barometer 
of  prices,  ought  not  to  be  affected  by  irrelevant  circum- 
stances, such  as  whether  the  price  of  cotton  is  quoted  by 
the  pound  or  by  the  bale.  Formula  51  alone  of  all  the  28 
formulae  will  be  so  affected  and  is  therefore  unreliable, 
very  erratic,  or  freakish.1  Finally,  every  other  simple  index 
number  may  be  considered  somewhat  freakish  because  its 
weights  are  arbitrarily  equal,  in  defiance  of  the  obvious 
inequalities  among  the  commodities  in  real  importance. 

Thus,  out  of  the  28  formulae,  we  know  that  18  are  biased. 
Of  these  18,  ten  are  also  freakish  (viz.  1,  11,  33,  35,  37, 
39,  43,  45,  47,  49).  Besides  these  there  are  four  other 
freakish  formulae  (viz.  21,  31,  41,  51).  This  leaves  only 
two  formulae  not  condemned  on  either  score.  These  two 
are  53  and  59  (or  3  and  5,  or  17  and  19). 

Formulae  53  and  59  are  very  close  together.  Thus,  al- 
ready, we  find  that  all  of  the  28  formulae  which  differ 
widely  from  each  other  have  a  discernible  reason  to  differ 
—  bias  or  freakishness  —  while  those  for  which  we  can- 
not discover  any  reason  for  differing  do  not,  in  fact,  differ 
very  much. 

§  15.  Bias  and  Errors  Generally  are  Relative 

We  shall  see  that  the  " ideal"  formula,  353,  gives  an 
almost  absolute  standard  by  which  to  measure  errors. 

1  This  feature  is  discussed  in  detail  in  Appendix  III. 


BIASED  INDEX  NUMBERS  117 

But,  for  the  present,  it  is  better  not  to  try  to  imagine 
any  absolute  standard,  however  much  we  may  dislike 
to  rest  on  mere  "relativity."  When  we  say,  for  instance, 
that  Formula  1  has  an  upward  bias  and  11  a  downward 
bias,  both  of,  say,  four  per  cent  in  1917,  we  mean  simply 
that  these  four  per  cent  errors  apply  in  addition  to  any 
other  errors  there  may  be.  We  thus  think  of  each  bias  as 
measured  relatively  to  the  half-way  point  between  1  and 
11,  but  without  assuming  necessarily  that  this  half-way 
point  is  itself  correct.  This  half-way  point  may,  for  aught 
we  yet  know,  be  too  high  by  ten  per  cent ;  in  which  case 
the  error  of  1  is  10  +  4,  or  14  per  cent,  and  of  11,  10  —  4, 
or  six  per  cent.  In  that  case  the  bias  of  11  is  still  four 
per  cent  downward,  despite  the  fact  that  the  net  error  is 
six  per  cent  in  the  opposite  direction.  Thus  we  may 
say,  as  compared  with  any  other  index  number  without 
assignable  bias,  Formula  1  has  an  upward  bias,  "other 
things  being  equal." 

§  16.  Historical 

The  term  "bias"  has  been  used  by  Bowley  and  other 
statisticians  as  applied  to  errors.  The  idea  of  type  bias 
was  expressed,  in  other  language,  by  Walsh.1  Also,  while 
he  did  not  recognize  weight  bias,  he  did  point  out  that  the 
arithmetic  average  should  be  used  with  the  weighting 
of  the  base  year  and  the  harmonic  with  the  weighting 
of  the  given  year.2 

Perhaps,  as  pointed  out  to  me  by  Walsh,  Sauerbeck 
had  an  inkling  of  the  upward  bias  of  the  arithmetic  aver- 
age in  a  passage  quoted  by  N.  G.  Pierson,3  although  Sau- 
erbeck had  no  remedy  to  propose. 

1  Measurement  of  General  Exchange  Value,  pp.  327-28. 

2  Ibid.,  pp.  307,  349. 

9  Economic  Journal,  March,  1896,  p.  128. 


CHAPTER  VI 

THE  TWO  REVERSAL  TESTS  AS  FINDERS  OF  FORMULA 

§  1.  The  Time  Reversal  Test  as  Finder  of  Formulae 

NOT  only  do  the  two  tests  reveal  joint  errors  per- 
taining to  each  formula,  but  they  afford  the  means  of 
rectification.  But  before  we  can  thus  rectify  any  given 
formula  we  must  first  find  for  it  two  other  formulae  re- 
lated to  it.  These  two  other  formulae  are  "antithetical" 
to  the  original  formulae;  one  being  its  antithesis  re- 
specting Test  1  and  the  other  its  antithesis  respecting 
Test  2.  These  two  antitheses  of  any  formula  will  there- 
fore be  called  its  time  antithesis  and  its  factor  antithesis. 
To  find  these  two  antitheses  is  our  next  task  and  the  ob- 
ject of  this  chapter. 

The  time  antithesis  of  any  given  formula  is  found  by 
applying  Test  1  to  that  formula.  As  we  know,  Test  1 
involves  two  steps : 

(1)  Interchanging  the  two  times  and  thus  obtaining 
the  index  number  reversed  in  time. 

(2)  Dividing  the  last  found  expression  into  unity. 
The  result  ought  to  be  the  original  formula  itself  in 

order  that  Test  1  may  be  fulfilled.  If  it  is  not,  then  the 
resulting  formula,  instead  of  being  identical  with  the 
original  formula,  is  its  time  antithesis.  That  is,  the  time 
antithesis  of  any  index  number  between  one  time  and 
another  is  found  by  applying  the  very  same  formula  the 
other  way  round  and  then  turning  it  upside  down. 

Algebraically,  the  first  step,  applying  the  formula  the 
other  way  round,  consists  in  interchanging  the  subscripts 

118 


TESTS  AS  FINDERS  OF  FORMULAE  119 

(say  "0"  and  "!")»  i.e.  erasing  "0"  wherever  it  occurs 
and  writing  "1"  in  its  place,  and  vice  versa.  Thus  For- 
mula 7,  viz., 


**(£) 


„ 

becomes 


The  second  step,  inverting,  i.e.  dividing  into  unity,  con- 
sists of  interchanging  numerator  and  denominator. 
Thereby,  the  above  becomes  the  required  time  antithesis, 


We  have  taken  a  particular  case  for  the  sake  of  illustra- 
tion. In  the  most  general  terms  the  process  is :  Let  P0i 
represent  any  index  number  for  time  "1"  relatively  to 
time  "0."  The  "other  way  round"  is  Pi0,  and  this 

"turned  upside  down"  is  — ,  which,   therefore,  is  the 

*  10 

general  expression  for  the  time  antithesis  of  P0i- 

It  may  easily  be  shown  that  the  antithetical  relation- 
ship is  necessarily  mutual,  the  original  formula  being  deriv- 
able by  the  very  same  process  from  its  antithesis,  so  that 
each  of  the  two  is  the  time  antithesis  of  the  other.1 

§  2.  Numerical  Illustration  of  Time  Antithesis 

Let  us  illustrate  these  two  steps  by  starting  once  more 
with  the  simple  arithmetic  index  number  of  prices  for 
1918  relatively  to  1917  and  repeating,  in  slightly  different 
form,  some  of  what  was  shown  in  Chapter  V.  This 
is  110.11  per  cent.  The  first  of  the  two  steps  is  to  inter- 
change 1917  and  1918,  i.e.  to  calculate  the  simple  arith- 
metic index  number  of  prices  for  1917  relatively  to  1918. 

i  See  Appendix  I  (Note  to  Chapter  VI,  §  1). 


120          THE  MAKING  OF  INDEX  NUMBERS 

This  is  94.46  per  cent.  But  these  two  index  numbers, 
forward  and  backward,  are  mutually  inconsistent,  not 
being  reciprocals,  i.e.  their  product  not  being  unity  or 
100  per  cent.  Test  1  is  not  fulfilled. 

But,  by  the  second  step,  we  divide  one  of  them,  94.46 
per  cent,  into  100  per  cent,  or  unity,  obtaining  105.86 
per  cent,  which  is  the  time  antithesis  of  110.11.  This 
105.86  per  cent  is  the  figure  which  multiplied  by  the  arith- 
metic backward,  94.46  per  cent,  will  give  the  true  required 
100  per  cent.  It  will  be  recognized  as  the  simple  harmonic. 

§  3.   Graphic  Illustration  of  Time  Antithesis 

Thus  the  simple  harmonic  is  the  time  antithesis  of  the 
simple  arithmetic.  The  illustration  of  this  relationship 

is   given   in  Chart  22. 
The  Harmon fc  forward  is  parallel  Here  the  original  index 

to  the  Arithmetic  backward.  number  is  that  of  the  36 

prices  as  they  changed 

from  1917  to  1918  and 
is  represented  by  the 
arithmetic  "forward" 
curve.  The  effect  on 
this  index  number  of 
interchanging  the  "O's" 

-  ./Q  and   "Ts,"   or  in  this 

CHABT22.    As  in  Charts  HP  and  \4Q  case>  interchanging  the 

the  harmonic  may  be  regarded  as  the  arith-  "4V   and  "5*8,     i.e.  in- 

in  disguise,  being  parallel  terchanging  the  years 
1917  and  1918,  is  repre- 
sented by  the  "backward"  pointing  curve.  This  shows 
how  the  simple  arithmetic  would  portray  the  price 
change  in  going  backward  from  1918  to  1917.  The  result 
is  represented  by  drawing,  parallel  to  the  last  named,  the 
dotted  line  pointing  forward-  This  is  the  harmonic. 


TESTS  AS  FINDERS  OF  FORMULAE  121 

§  4.  Algebraic  Expression  of  Arithmetic  and  Harmonic 
Time  Antitheses 

Beginning  with  Formula  1  (simple  arithmetic)  and 
subjecting  it  to  our  twofold  procedure  we  obtain : 

Original  Formula  1 

(1)  Interchanging  the 

(2)  Inverting  -r-p- 

2(2.°) 
W 

The  result  is  the  time  antithesis  of  the  original  simple 
arithmetic.  But  the  formula  thus  found  is  evidently 
Formula  11,  the  " simple  harmonic."  That  is,  Formulae  1 
and  11  are,  as  noted  in  the  last  section,  time  antitheses 
of  each  other.  The  harmonic  (forward)  is  thus  the  arith- 
metic backward  reversed  in  direction.1 

Next  take  the  arithmetic  weighted  7. 


<*) 

"O's"  and "IV— — 


Original  Formula  3 

(1)  Interchanging  the  "O's"  and  "  1's" 

(2)  Inverting 

,pi> 

which  is  Formula   19,   harmonic   weighted   IV.    Thus 
Formulae  3  and  19  are  time  antitheses. 

1  Cf.  C.  M.  Walsh,  Measurement  of  General  Exchange  Value,  pp.  327-28. 


122 


THE  MAKING   OF  INDEX  NUMBERS 


Similarly,  Arithmetic  Formula  5  and  Harmonic  Formula  17  are  time 
antitheses. 

Similarly,  Arithmetic  Formula  7  and  Harmonic  Formula  15  are  time 
antitheses. 

Similarly,  Arithmetic  Formula  9  and  Harmonic  Formula  13  are  time 
antitheses. 

Tabulating  more  simply,  we  may  indicate  the  time 
antitheses  by  connecting  lines  as  follows,  the  weighted 
arithmetic  being  related  to  the  weighted  harmonic  in 
reverse  order: 


WEIGHTING 

ARITHMETIC 

HARMONIC 

Formula  No. 

Fonnula  No. 

Simple 

1<n,_ 

.    >  11 

Weighted     7 

3* 

y13 

Weighted   II 

5«*^\ 

X^15 

Weighted  III 

7-*-~"/ 

'N^^IT 

Weighted  IV 

3i  19 

§  5.  Time  Antithetical  Geometries,  Medians,  and 
Modes 

The  other  formulae  in  our  list  are  also  related. 
Algebraically,  applying  the  same  processes  to  the  sim- 
ple geometric,  we  have  : 


Original  Formula  21 


p0 


p    o 


"f/>\    ^'.A    /p''<A 
X  W    Vi/    VV 


(1)  Interchanging    the    " 

"O's"  and  "  1's" 

(2)  Inverting    (and    sim-   "/fpAv/'PY\v/'P"1W 

plif  ying)  V  (-J  X  (yj  X  y^-J  X 

which  is  identical  with  the  original  formula.    Thus  Test  1 


TESTS  AS  FINDERS  OF  FORMULAE 


123 


is  fulfilled  or,  if  we  wish  to  say  so,  the  simple  geometric 
is  its  own  time  antithesis. 

Likewise,  taking  the  geometric  weighted  7,  we  find 
its  time  antithesis  to  be  geometric  weighted  IV  while, 
similarly,  geometric  II  and  geometric  III  are  the  time 
antitheses  of  each  other  so  that  the  antithetical  relation- 
ships of  the  geometries,  as  shown  by  connecting  lines,  are  : 


Simple  Geometric 


Weighted  / 
Weighted  II 
Weighted  III 
Weighted  IV 


21) 


Likewise  the  simple  median,1  Formula  31,  fulfills  Test  1 
and  the  weighted  medians  have  the  same  sort  of  relation- 
ships as  in  the  case  of  the  geometries,  i.e. 


33 
35 
37 
39 


The  same  scheme  of  relationships  applies  to  the  mode. 


43 

45} 
47} 
49 


1  If  the  number  of  terms  is  even,  the  test  is  fulfilled  only  provided  we 
take  the  geometric  mean  of  the  adjacent  middle  terms  as  the  median. 


124          THE  MAKING   OF  INDEX  NUMBERS 

§  6.  Antithetical  Aggregatives 

The  simple  aggregative  comes  next. 

Original  formula  ^^ 

(1)  Interchanging  the  "0V  and  "IV  ^-° 

2.pi 

(2)  Inverting  J^ 

Po 

which  is  identical  with  the  original,  Formula  51. 
Taking  Formula  53,  the  aggregative  weighted  7,  we  have 

Original  Formula  53 

(1)  Interchanging  the  "O's"  and  "IV 

(2)  Inverting 

which  is  not  identical  with  the  original  (53)  but  is  For- 
mula 59. 
Thus  we  have 

51) 

631 

59 


§  7.  Review  of  the  28  Formulae 

We  have  now  paired  as  time  antitheses  all  our  28  for- 
mulse  relatively  to  Test  1.  Evidently,  if  we  had  started 
with  only  one  member  of  each  pair,  we  could  have  dis- 
covered the  other  member  by  means  of  the  twofold  pro- 
cedure. For  instance,  if  we  had  not  included  the  har- 


TESTS  AS  FINDERS  OF  FORMULAE  125 

monies  in  our  original  list,  we  should  have  been  led  to 
discover  them  by  applying  the  twofold  procedure  to  the 
arithmetics,  or  vice  versa.  Again,  if  we  had  not  included 
in  our  original  list  the  weighted  formulae  whose  identi- 
fication numbers  end  in  7  and  9,  we  should  have  been  led  to 
discover  them,  by  applying  the  twofold  procedure  to  those 
whose  identification  numbers  end  in  3  and  5,  or  vice  versa. 

§  8.  The  Factor  Reversal  Test  as  Finder  of  Formulae 

Thus  far  we  have  considered  only  Test  1  (time  reversal 
test)  as  a  finder  of  formulae ;  and  the  formulae  we  found 
were  formulae  already  discussed  —  a  harmonic  for  an 
arithmetic,  a  weighted  geometric  for  a  differently  weighted 
geometric,  etc.  The  test  was  to  reverse  the  times  and 
then  invert  the  resulting  formula  (i.e.  divide  it  into  unity). 
This  was  called  the  time  antithesis,  and  turned  out  in 
every  case  to  be  an  old  formula. 

When  we  apply  Test  2  (the  factor  reversal  test),  how- 
ever, we  shall  actually  be  led  to  new  formulae  not  included 
in  the  previous  28. 

The  factor  antithesis  of  any  given  formula  for,  say, 
the  price  index,  is  found  by  applying  Test  2  to  that  for- 
mula. Test  2  involves  two  steps : 

(1)  Interchanging  the  prices  and  quantities,  thus  ob- 
taining the  index  number  of  quantities. 

(2)  Dividing  the  last  found  expression  into  the  value  ratio . 
The  result  ought  to  be  the  original  formula  itself  in 

order  that  Test  2  may  be  fulfilled.  If  it  is  not,  then  the 
resulting  formula,  instead  of  being  identical  with  the 
original  formula,  is  its  factor  antithesis. 

§  9.  Numerical  Illustration  of  Factor  Antithesis 

Consider  the  year  1917.  The  simple  arithmetic  index 
number  of  prices  for  1917  is  175.79  per  cent  of  the  1913 


126          THE  MAKING  OF  INDEX  NUMBERS 


f 


Three  Types  of  Index  Numbers  /      ,22 

of  Prices  ' 


Factor  Antitheses  of 

Harmonic 


// 


Arifhmeffc 


v 

^ 


-^ 


XJ  '/4  75  '16  '17  1$ 

CHART  23P.  In  the  case  of  the  factor  antitheses,  the  harmonic  is  above 
and  the  arithmetic  below  the  geometric,  in  an  order  the  reverse  of  that  in 
which  the  original  three  types  were  arrayed. 

price  level,  and  the  corresponding  index  number  of  quan- 
tities for  1917  is  125.84  per  cent  of  the  1913  quantity  level. 
One  of  these  is  too  big,  since  their  product  evidently  ex- 
ceeds the  value  ratio,  which  is  only  192.23  per  cent.  By 


TESTS  AS  FINDERS  OF  FORMULA  127 

dividing  this  value  ratio,  192.23,  by  the  second  factor 
(quantity  index),  125.84,  we  get  a  new  price  index,  152.76. 
This  we  call  the  factor  antithesis  of  the  original  price 
index,  175.79  —  that  is,  the  factor  antithesis  of  the  simple 

Three  Types  of  Index  Numbers 
of  Quantities 

Factor  Antitheses  of 

Harmonic 

Geometric  ^ -—- 

Arithmetic  _—•"""""        - 


VJ  14  IS  'Iff  '17  18 

CHABT  23Q.    Analogous  to  Chart  23P. 

arithmetic  index  of  prices.    It  is  the  figure  which,  when 
used  as  a  factor  and  multiplied  by  the  simple  arithmetic 
quantity  index,  will  give  the  true  required  192.23  per 
cent. 
Or,  reversely,  dividing  the  value  ratio,  192.23  per  cent, 


128         THE  MAKING  OF  INDEX  NUMBERS 

by  the  first  factor  (the  price  index),  175.79  per  cent,  we 
obtain  109.35  per  cent,  the  factor  antithesis  of  the  quan- 
tity index,  125.84  per  cent,  i.e.  the  figure  which  if  used 

Four  Methods  of  Weighting  Compared 
(Prices) 


73  H  '/5  IB  '17  '/9 

CHART  24P.  Comparison  of  methods  of  weighting  applied  to  factor 
antitheses  of  the  index  numbers  given  in  Chart  16P.  The  change  from 
weights  with  base  prices  (curve  numbers  ending  in  4  and  6)  to  weights  with 
given  year  prices  (curve  numbers  ending  in  8  and  0)  shifts  the  curve  down- 
ward. Changes  in  the  quantities  have  little  effect  one  way  or  the  other. 

as  the  quantity  factor  and  multiplied  by  the  original  price 
factor,  175.79,  will  give  the  true  value  ratio,  192.23  per  cent. 
The  factor  antithesis  of  Formula  1  is  numbered  2 ; 
the  factor  antithesis  of  Formula  3  is  numbered  4,  and 
so  on.  That  is,  each  odd  numbered  formula  has  as  its 
factor  antithesis  the  following  even  number.1 

1  The  complete  system  of  numbering  formulae  is  given  in  Appendix  V,  §  2. 


TESTS  AS  FINDERS  OF  FORMULAE  129 

§  10.   Graphic  Illustration  of  Factor  Antithesis 

Graphically,  Charts  23P  and  23(?  show  three  principal 
types  of  factor  antitheses  arranged  in  five  groups  by 
weights.  The  order  is  in  each  case  the  reverse  of  that  of 
the  original  index  number  (see  Charts  15P  and  15Q).  The 

Four  Methods  of  Weighting  Compared 

(Quantities) 


V3  '#  '/5  V^  77  18 

CHART  24Q.  Analogous  to  Chart  24P.  The  shifts  are  equal  (and  oppo- 
site) to  those  of  16P.  Those  of  24P  are  equal  and  opposite  to  those  of  16Q. 

factor  antitheses  (even  numbered)  of  the  price  indexes 
exhibit  the  same  biases  as  the  original  quantity  indexes 
(odd  numbered)  in  the  reverse  order. 

Charts  24P  and  24Q,  classifying  the  opposite  way, 
show  the  four  varieties  of  factor  antitheses  correspond- 
ing to  the  four  systems  of  weighting,  arranged  in  three 
groups  by  types. 


130         THE  MAKING  OF   INDEX  NUMBERS 

Charts  20P  and  20Q,  lower  part,  show  the  combination 
diagram  for  the  factor  antitheses.  It  is  similar  to 
Charts  20P  and  20Q,  upper  part,  in  reverse  order. 

Charts  17 P  and  17Q,  lower  part,  show  the  factor  antith- 
eses of  the  median.  They  differ  only  slightly  from  each 
other  and  exhibit  the  same  inertness  or  tendency  for  34 
and  36  (and  38  and  40)  to  stick  close  together,  except 
occasionally  when  they  fly  apart. 

The  factor  antitheses  of  the  modes  (not  charted)  would 
be  indistinguishable  from  each  other. 

§  11.  Algebraic  Expression  of  Factor  Antitheses 

Algebraically,  the  first  step,  interchanging  prices  and 
quantities,  consists  merely  in  interchanging  "p"  and 
"q"  in  any  formula,  i.e.  erasing  "p"  wherever  it  occurs, 
and  writing  "q"  in  its  place,  and  vice  versa;  the  second 
step  is  dividing  the  result  into  the  value  ratio. 

For  example,  according  to  the  simple  arithmetic,  the 
index  number  for  time  1  relatively  to  time  0  is 


Original  Formula  1 

2 

(1)  Interchanging  "p's"  and  "g's" 

2 

/iTIifl*  ^ 

(2)  Dividing  into 


n 

Thus  Formula  1  does  not  meet  Test  2,  but  the  appli- 
cation of  that  test  leads  to  a  new  formula,  the  factor  antith- 
esis of  the  former. 


TESTS  AS  FINDERS  OF  FORMULAE  131 


Again,  Formula  7,  viz.,     y      —  becomes 


The  second  step  consists  in  dividing  the  last  found  into 
,  giving  Formula  8. 

The  above  are  particular  cases  for  illustration.  In 
the  most  general  terms  we  may  let  P0i  be  any  index 
number  for  prices.  Substituting  "q's"  for  "p's,"  and 
vice  versa,  we  get  Q0i,  and  dividing  into  the  value  ratio, 

we  get—  "Q--i-  Qoi  as  the  general  expression  for  the  fac- 


tor  antithesis  of  P0i. 

The  (even  numbered)  weighted  antitheses  of  price 
indexes  exhibit  the  same  biases  as  the  (odd  numbered) 
original  quantity  indexes  in  opposite  order. 

§  12.   The  Various  Roles  of  Laspeyres'  and 
Paasche's  Formulae 

In  precisely  the  same  way  we  may  obtain  all  the  other 
factor  antitheses.  In  the  case  of  Formula  3  (or  its  substi- 
tutes, Formulae  17  and  53)  the  result  is  subject  to  simplifica- 
tion. 

As  we  already  know,  Formula  3  reduces  to  53.  Let 
us  start  therewith  and  apply  the  factor  reversal  test. 

Original  Formula  53 


(1)  Interchanging  "p's"  and  "g's" 

(2)  Dividing  this  into^^-1  and  canceling  Sp0g0 


which  (according  to  our  identification  numbering)  is  called 
Formula  54.    This  is  evidently  identical  with  Formula  59. 


132         THE  MAKING  OF  INDEX  NUMBERS 


Thus  it  will  be  noted  that  the  factor  antithesis  of  For- 
mula 53  (namely  54)  is  identical  with  its  time  antithesis, 
59,  which  we  have  known  as  Paasche's  formula.  Here- 
after Paasche's  formula  will  be  usually  referred  to  as  54 
rather  than  as  59. 

For  the  sake  of  uniformity  of  method  we  designate 
the  factor  antithesis  of  Formula  53  as  54  (and  likewise 
of  Formula  3  as  4,  and  of  Formula  17  as  18,  all  of  which 
[54,  4,  18]  are  identical  with  59).  Again,  starting  with 
Formula  59,  we  get  as  its  factor  antithesis  a  formula  desig- 
nated as  60  which,  of  course,  turns  out  to  be  identical 
with  53.  Likewise  the  factor  antitheses  of  Formulae  5 
(called  6)  and  of  19  (called  20)  are  all  identical  with  60. 

Our  table  of  formulae  now  has  two  sets  of  six  identicals 
(3,6,  17,  20,  53,  60)  and  (4,5,18,  19,54,59),  represent- 
ing two  types,  type  L  (Laspeyres')  and  type  P  (Paasche's). 

The  following  list  of  weighted  arithmetic,  harmonic, 
and  aggregative  formulae  is  arranged  to  show  the  repeti- 
tions of  L  (Laspeyres')  and  P  (PaascnVs)  formulae  (the 
only  repeaters  in  the  entire  list  of  formulae). 


ARITHMETIC 

AGGREGATIVE 

HARMONIC 

Formula  No. 

Formula  No. 

Formula  No. 

13 

14 

15 

16 

3L 

53  L 

17  L 

4P 

54  P 

18  P 

5P 

59  P 

19  P 

6L 

60  L 

20  L 

7 

8 

9 

10 

Thus  L  and  P  fall  always  among  these  three  types. 


TESTS  AS  FINDERS  OF  FORMULAE 
§  13.  List  of  46  Formulae 


133 


We  had  28  formulae  which  included  four  identicals. 
All  the  even  numbered  formulae  which  we  have  just  added 
to  the  list  by  applying  Test  2  are  new,  excepting  only 
Formulae  54  and  60  and  their  identicals.  Thus  instead 
of  28  formulae,  or  24  after  canceling  identicals,  we  now 
have  56,  or  46  after  canceling  identicals.  They  are  as 
follows : 


ARITHMETIC 

HARMONIC 

GEOMETRIC 

MEDIAN 

MODE 

AGGREGATIVE 

1 

11 

21 

31 

41 

51 

2 

12 

22 

32 

42 

52 

3L 

13 

23 

33 

43 

53  L 

4P 

14 

24 

34 

44 

54  P 

5P 

15 

25 

35 

45 

6L 

16 

26 

36 

46 

7 

17  L 

27 

37 

47 

8 

18  P 

28 

38 

48 

9 

19  P 

29 

39 

49 

59  P 

10 

20  L 

30 

40 

50 

60  L 

The  following  list  omits  duplicates  (53  and  54  being 
retained  but  their  duplicates  omitted). 


u 

| 

W 

GEOMETRIC 

MEDIAN 

J 

| 

Simple    

1 

11 

21 

31 

41 

r:i 

Fac.  an.  of  simple  

2 

12 

22 

32 

42 

52 

Weighted  7    
Fac.  an.  of  weighted  7  ... 
Weighted  77   

- 

13 
14 
15 

23 
24 
25 

33 
34 
35 

43 
44 
45 

53 
54 

Fac.  an.  of  weighted  77  .  . 
Weighted  777   

7 

16 

26 
27 

36 
37 

46 
47 

Fac.  an.  of  weighted  777.  . 
Weighted  IV  

8 
9 

- 

28 
29 

38 
39 

48 
49 

Fac.  an.  of  weighted  IV.  . 

10 

- 

30 

40 

50 

- 

134         THE  MAKING  OF  INDEX  NUMBERS 

These  46  formulae  may  be  called  the  primary  formulae. 
The  additional  ones  which  follow  in  subsequent  chapters 
are  all  derivatives  from  these  46  primary  formulae. 

Of  these  46  distinct  formulae : 

six  are  simples,  viz.  1,  11,  21,  31,  41,  51 

six  are  the  factor  antitheses  of  the  simples,  viz.  2,  12,  22,  32,  42,  52 

two  are  Laspeyres'  and  Paasche's,  53,  54 

(aggregatives  which  interchange  with  some  of 

the  arithmetics  and  harmonics) 

two  are  other  weighted  arithmetics,  7,     9 

two  are  the  factor  antitheses  of  these,  8,  10 

two  are  other  weighted  harmonics,  13,  15 

two  are  the  factor  antitheses  of  these,  14,  16 

four  are  weighted  geometries,  23,  25,  27,  29 

four  are  the  factor  antitheses  of  these,  24,  26,  28,  30 

four  are  weighted  medians,  33,  35,  37,  39 

four  are  the  factor  antitheses  of  these,  34,  36,  38,  40 

four  are  weighted  modes,  43,  45,  47,  49 

four  are  the  factor  antitheses  of  these,  44,  46,  48,  50 

§  14.   Historical 

As  has  been  pointed  out  in  previous  chapters,  the  time 
reversal  test  has,  to  all  intents  and  purposes,  been  used 
by  many  previous  writers.  These  same  writers,  notably 
Walsh,  have  likewise  observed  the  essential  symmetry 
of  Fdrfnulae  23  and  29,  of  1  and  11,  and  of  3  and  19  (or  53 
and  59). 

As  to  factor  antitheses,  Formula  52  has  been  used  by 
Drobisch  and  Sir  Rawson-Rawson  (who  proposed  to 
measure  the  average  price  level  of  imports  or  exports  by 
dividing  values  by  tonnage).  Formula  22  has  been  pro- 
posed by  Nicholson  and  Walsh.  Among  other  factor 
antitheses  Formula  2154  (to  be  described  later)  was 
proposed  by  Walsh  while  4154  (also  to  be  described 
later)  has  been  proposed  by  Lehr.  As  these  are  all  factor 
antitheses  of  other  formulae,  the  principle  of  such  antith- 
esis must  have  been  more  or  less  consciously  recognized. 
The  other  factor  antitheses  in  our  list,  derived  from  the 


TESTS  AS  FINDERS  OF  FORMULA  135 

general  application  of  the  principle,  seem  not  to  have 
been  expressed.  Nevertheless  the  general  principle  may 
be  said  to  be  recognized  whenever  any  series  of  statistics 
of  money  values  (such  as  of  imports,  exports,  production, 
clearings)  are  " deflated"  by  dividing  by  an  index  number 
of  prices  to  obtain  a  rough  index  of  the  underlying  quanti- 
ties (physical  volume  of  imports,  exports,  production, 
trade). 


CHAPTER  VII 
RECTIFYING  FORMULAE  BY  "CROSSING"  THEM 

§  1.  Crossing  Time  Antitheses 

WE  have  thus  far  reached  two  chief  results  from  the 
use  of  the  tests.  First,  we  have  noted  which  formulae 
meet,  and  which  fail  to  meet,  these  tests.  Of  the  46, 
only  four,  the  simple  geometric,  median,  mode,  and  aggre- 
gative, meet  Test  1  and  none  of  the  46  meet  Test  2.  Sec- 
ondly, we  have,  by  Test  1,  found  for  each  formula  its 
time  antithesis  (in  each  case  an  old  odd  numbered  for- 
mula) and  by  Test  2  its  factor  antithesis  (in  each  case, 
except  for  some  duplications,  a  new  even  numbered  for- 
mula). 

We  now  come  to  a  third  use  of  these  tests,  namely,  to 
" rectify"  formulae,  i.e.  to  derive  from  any  given  formula 
which  does  not  satisfy  a  test  another  formula  which  does 
satisfy  it ;  so  that  now  we  are  about  to  pass  from  our 
46  primary  formulae  to  the  region  of  derivative  formulae. 

This  is  easily  done  by  "crossing,"  that  is,  by  averaging, 
antitheses.  If  a  given  formula  fails  to  satisfy  Test  1 
its  time  antithesis  will  also  fail  to  satisfy  it ;  but  the  two 
will  fail,  as  it  were,  in  opposite  ways,  so  that  a  cross  be- 
tween them  (obtained  by  geometrical  averaging)  will 
give  the  golden  mean  which  does  satisfy.  This  will  be 
true  in  all  cases,  whether  the  formulae  paired  and  crossed 
are  arithmetic,  harmonic,  geometric,  median,  mode,  or 
aggregative  (or  any  other  for  that  matter).  As  will  be 
shown,  the  geometric  mean  of  the  two  antithetical  index 
numbers  may  always  be  used  for  "crossing"  them 

136 


RECTIFYING  FORMULA  BY  "CROSSING"     137 

whether  the  two  themselves  be  geometric,  median,  mode, 
aggregative,  or  one  arithmetic  and  the  other  harmonic 
(arithmetic-harmonic).  If  we  thus  cross  the  two  antithe- 
ses geometrically,  the  resulting  formula  will  satisfy 
the  test.  But  if  we  cross  them  arithmetically,  or  har- 
monically, it  will  not.  By  this  simple  process  of  crossing 
(geometrically)  we  can  " rectify"  any  formula  whatever 
so  far  as  securing  conformity  to  either  or  both  of  the  two 
tests  goes. 

Thus,  take  the  simple  arithmetic.  Its  time  antithesis 
is  the  simple  harmonic.  Neither  of  these  fulfills  the  first 
or  time  reversal  test.  But  the  failure  of  each  in  one  di- 
rection is  exactly  matched  by  the  failure  of  the  other  in 
the  opposite  direction,  and  we  shall  see  that  the  cross 
between  the  two  meets  the  test  exactly. 

§  2.  Numerical  Illustration 

The  simple  arithmetic  index  number  for  1917  on  1913 
as  base  is  175.79  per  cent.  The  simple  harmonic  (time 
antithesis)  for  1917  on  1913  as  base  is  157.88  per  cent. 
Neither  of  these  satisfies  Test  1  but  the  cross  between 
them  is  166.60  per  cent  which  does  satisfy  Test  1  since  it 
is  the  reciprocal  of  60.02  per  cent,  the  figure  reached  by 
the  same  process  applied  the  other  way  round  in  time. 

Thus  166.60,  the  rectified  arithmetic  (and,  of  course, 
the  rectified  harmonic  as  well),  unlike  the  original  un- 
rectified  or  simple  arithmetic,  175.79  per  cent,  and  the 
original  unrectified  or  simple  harmonic,  157.88,  conforms 
to  Test  1,  i.e.  is  such  that  multiplied  by  the  similarly 
obtained  figure  for  the  reverse  direction,  60.02  per  cent, 
it  gives  exactly  100  per  cent,  or  unity;  in  other  words, 
the  forward  and  backward  are  reciprocals. 

The  simple  geometric  index  number,  on  the  other  hand, 
being  its  own  time  antithesis,  i.e.  conforming  to  Test  1, 


138 


THE  MAKING  OF  INDEX  NUMBERS 


requires  no  rectification  (so  far  as  that  test  goes).  For 
1917  this  simple  geometric  index  number  is  166.65  per 
cent  and,  in  the  reversed  direction,  it  is  60.01  per  cent, 
which  is  the  reciprocal  of  166.65  per  cent. 


The  Simple  Geometric 
compared  with  the  simple  Arithmetic  and 
Harmonic  and  their  reef  if /cat/on  by  test  I. 
(Prices) 


73  14  75  76  77  78 

CHAKT  25  P.  The  geometric  (21)  is  practically  identical  with  101,  the 
geometric  mean  of  1  and  11. 

The  entire  (price)  series  of  the  two,  i.e.  the  recti- 
fied simple  arithmetic-harmonic,  101,  and  the  simple 
geometric,  21,  are : 


1913 

1914 

1915 

1916 

1917 

1918 

Rectified  Arithmetic- 
Harmonic  (101) 

100 

95.75 

96.80 

121.38 

166.60 
166.65 

179.09 
180.12 

Simple  Geometric  (21) 

100 

95.77 

96.79 

121.37 

Comparison  between  these  two  index  numbers  satisfy- 
ing Test  1  reveals  an  unexpected  result  —  that  is,  a  re- 
markably close  agreement.  Thus,  the  supposed  conflict 
between  the  geometric  and  arithmetic  index  numbers 
disappears  by  " rectification." 


RECTIFYING  FORMULAE  BY  "CROSSING"     139 

Hitherto  there  has  been  a  disposition  to  think  that  the 
arithmetic  and  geometric  stand  on  an  equality,  that,  while 
the  arithmetic  lies  above  the  geometric  and  the  harmonic 
lies  below  it,  this  is  little  more  than  an  interesting  fact. 

Jevons  and  a  few  others,  on  the  other  hand,  have  had 
a  disposition  to  prefer  the  geometric  as  one  always  pre- 
fers a  " golden  mean"  to  extremes,  but  without  assigning 
any  clear  reason  for  the  preference.  The  mere  fact  that 
the  geometric  lies  between  two  others  is  not  a  very 
logical  reason  for  preferring  it. 

The  Simple  Geometric 

compared  with  the  simple  Arithmetic  and 
Harmonic   and  their  rectification  by  test  f 
(Quantities) 


73  '14  75  16  17  18 

CHART  25Q.      Analogous  to  Chart  25P.     But  21  and  101  disagree  in  1918. 

We  did,  however,  find  a  very  good  reason  for  rejecting 
the  simple  arithmetic  (and,  likewise,  the  simple  harmonic) 
index  number.  It  will  not  work  both  ways  in  time  con- 
sistently with  itself.  But  when,  by  "  rectification,"  this 
defect  is  remedied  the  resulting  rectified  arithmetic  no 
longer  presents  any  problem  arising  from  results  dis- 
crepant with  the  geometric  mean. 

Test  1  thus  serves  as  a  touchstone  for  (1)  convicting 
the  arithmetic  (and  harmonic)  of  self -inconsistency ; 
(2)  remedying  that  inconsistency,  reaching  another  for- 
mula entirely  free  of  this  defect. 


140         THE  MAKING  OF  INDEX  NUMBERS 

§  3.   Graphic  Illustration 

Graphically,  Charts  25P  and  25Q  show  the  rectified  index 
number  (Formula  101)  by  crossing  the  simple  arithmetic  (1) 
and  simple  harmonic  (11)  and  its  practical  identity  with 
the  simple  geometric  (21). 

§  4.  Algebraic  Proof  that  Rectification  Can  Always 
be  Accomplished  by  Crossing  Time  Antitheses 

Algebraically,  the  full  proof  is  ridiculously  simple.  Let 
POI  be  any  formula  for  the  index  number  of  prices  of  date 
1  relatively  to  date  0.  Its  time  antithesis,  as  was  shown 

in  §  1  of  the  preceding  chapter,  is  —  -.     The  geometric 

PIO 
mean  is  found  by  multiplying  these  two  expressions  to- 

fiP~~ 

gether  and  extracting  the  square  root,  \^-      This  is 

"  10 

the  new  formula  which  we  are  to  prove  conforms  to 
Test  1. 

Let  us  apply  Test  1. 


(1)  Interchanging  the  "O's"  and  "1's,"  - 

POI 

(2)  Multiplying  this  by  the  original,  we  get  unity  as 
the  test  requires. 

Therefore  the  cross  between  any  two  tune  antitheses 
will  obey  Test  1. 

Thus,  as  Formulae  1  and  11  are  time  antitheses,  the 
formula  V(l)  X  (11)  must  fulfill  Test  1,  the  time  re- 
versal test.  We  have  called  this  new  formula,  101.  In 
all,  we  may  derive  the  following  new  formulae  fulfilling 
the  time  reversal  test  by  virtue  of  the  fact  that  each  is  a 
cross  between  two  time  antitheses  : 


RECTIFYING  FORMULAE  BY  "CROSSING"    141 

§  5.  List  of  Rectified  Formulae  by  Crossing  Time 
Antitheses 

Formulae  derived  from  arithmetics  and  harmonics. 
Vl  X  11  or  Formula  101 
V2  X  12  or  Formula  102 
V3  X  19  or  Formula  103 
V4  X  20  or  Formula  104 
V5  X  17  or  Formula  105 
V6  X  18  or  Formula  106 
V7  X  15  or  Formula  107 
Vg  X  16  or  Formula  108 
A/9  X  13  or  Formula  109 


VlO  X  14  or  Formula  110 
Formulae  derived  from  geometries : 

V23  X  29  or  Formula  123 
V24  X  30  or  Formula  124 
V25  X  27  or  Formula  125 
V26  X  28  or  Formula  126 
Formulae  derived  from  medians : 


V33  X  39  or  Formula  133 
-N/34  X  40  or  Formula  134 
A/35  X  37  or  Formula  135 
V36  X  38  or  Formula  136 
Formulae  derived  from  modes : 


V43X49  or  Formula  143 


V44  X  50  or  Formula  144 
V45  X  47  or  Formula  145 
V46  X  48  or  Formula  146 


142         THE  MAKING  OF  INDEX  NUMBERS 

Formulae  derived  from  aggregatives  : 

\/53  X  59  or  Formula  153 
V54  X  60  or  Formula  154 

Formula  153  for  prices  is : 


and  for  quantities : 

This  is  what  we  shall  call  our  " ideal"  formula.  It  is 
evidently  identical  with  Formula  154,  since  53  is  identi- 
cal with  60  and  59  with  54.  Likewise  the  resulting  153 
and  154  duplicate  Formulae  103,  104,  105,  106  (which 
result  from  various  identicals  of  53  and  54). 

In  numbering  these  rectified  formulae  for  identification, 
it  will  be  observed  that  we  simply  use  the  number  100,  in- 
creased by  the  number  of  the  lower  numbered  of  the  two 
time  antitheses  from  which  each  is  derived.1 

§  6.   Crossing  Factor  Antitheses 

We  come  now  to  Test  2,  and  factor  antitheses.  Recti- 
fication relatively  to  Test  2  is  accomplished  by  taking 
the  geometric  mean  between  any  two  formulae  which  are 
factor  antitheses.  Again  the  proof,  given  in  the  Appen- 
dix, is  simple.2 

§  7.  List  of  Rectified  Formulae  by  Crossing  Factor 

Antitheses 

Thus  we  obtain  the  following  formulae  conforming  to 
Test  2 : 

Vl  X    2  or  Formula  201 
V3  X    4  or  Formula  203 

1  The  complete  system  of  numbering  formula?  is  given  in  Appendix  V,  §  2. 

2  See  Appendix  I  (Note  to  Chapter  VII,  §  6)  for  proof  and  discussion. 


RECTIFYING  FORMULA  BY  "CROSSING"    143 

V5  X    6  or  Formula  205 

V?  X    8  or  Formula  207 

V9  X  10  or  Formula  209 
Vll  X  12  or  Formula  211 
Vl3  X  14  or  Formula  213 


V15  X  16  or  Formula  215 


Vl7  X  18  or  Formula  217 
Vl9  X  20  or  Formula  219 
V21  X  22  or  Formula  221 
V23  X  24  or  Formula  223 
V25  X  26  or  Formula  225 


v  27  X  28  or  Formula  227 
V29  X  30  or  Formula  229 
V31  X  32  or  Formula  231 
V33  X  34  or  Formula  233 
V35  X  36  or  Formula  235 
V37  X  38  or  Formula  237 
A/39  X  40  or  Formula  239 
A/41  X  42  or  Formula  241 
A/43  X  44  or  Formula  243 
A/45  X  46  or  Formula  245 
V47  X  48  or  Formula  247 
V49  X  50  or  Formula  249 
V51  X  52  or  Formula  251 
A/53  X  54  or  Formula  253 
V59  X  60  or  Formula  259 

In  numbering  these  formulae  for  identification,  it  will 
be  observed  that  we  simply  use  the  number  200,  increased 
by  the  number  of  the  lower  numbered  of  the  two  fader 


144         THE  MAKING  OF  INDEX  NUMBERS 

antitheses  from  which  each  is  derived  (just  as,  in  reference 
to  Test  1  we  used  the  number  100  plus  the  number  of 
the  lower  numbered  of  the  two  time  antitheses).  Of 
course,  the  ability  of  a  formula  to  conform  to  one  of  the 
two  tests  does  not  necessarily  imply  ability  to  conform 
to  the  other  (although,  as  a  matter  of  fact,  it  tends  in 
that  direction).  Accordingly,  most  of  the  200-group 
of  formulae  are  distinct  from  —  even  though  usually 
giving  results  close  to  —  the  100-group  of  formulae. 

Among  the  200-group  there  are  six  alike,  viz.,  those 
crossing  Laspeyres'  and  Paasche's,  or  the  following : 
Formulae  203,  205,  217,  219,  253,  259 ;  and  these  are  not 
only  identical  with  each  other  but,  as  will  be  seen  by 
inspection,  identical  with  six  of  the  100-group,  namely 
with  Formulae  153  and  154,  and  the  duplicates  of  the 
latter,  namely  Formulae  103, 104, 105,  106.  This  formula, 


mentioned  before  as  our  ideal,  is  the  cross  between  Las- 
peyres' and  Paasche's.  It  is  the  only  formula  which 
occurs  both  in  the  100  and  the  200  lists. 

§  8.  Fourfold  Relationship  of  Antitheses 

It  may  be  easily  shown1  that,lf  any  two  index  numbers 
are  time  antitheses  of  each  other,  then  their  respective 
factor  antitheses  are  also  time  antitheses  of  each  other. 
Thus,  Formulae  1  and  11  being  time  antitheses  of  each 
other,  Formulae  2  and  12  (their  factor  antitheses)  are 
also  time  antitheses  of  each  other.  Likewise  Formulae 
23  and  29  being  time  antitheses,  Formulae  24  and  30  (their 
respective  factor  antitheses)  are  also  tune  antitheses  of 
each  other. 

1  Algebraically,  the  proof  of  this  theorem  is  simple  and  is  given  in  Ap- 
pendix I  (Note  A  to  Chapter  VII,  §  8). 


RECTIFYING  FORMULAE  BY  "CROSSING"    145 

Similarly  it  may  be  easily  shown  1  that  if  any  two  in- 
dex numbers  are  factor  antitheses  of  each  other,  then  their 
respective  time  antitheses  are  also  factor  antitheses  of 
each  other. 

§  9.  Rectifying  Simple  Arithmetic  and  Harmonic 
by  Both  Tests 

Thus  we  find  our  formulae  arranging  themselves  in 
quartets,  which  not  only  form  two  pairs  of  time  antithe- 
ses, but  also  form  two  pairs  of  factor  antitheses  —  all  of 
them  failing  to  meet  tests,  but  rectifiable  through  cross- 
ing. 

Thus  the  quartet  of  formulae : 

1  11 

2  12 

are  such  that  either  horizontal  pair  yields  a  formula  con- 
forming to  Test  1  (i.e.  V  (1)  X  (11)  is  Formula  101,  and 
V  (2)  X  (12)  is  Formula  102)  while  the  vertical  pairs 
yield  formulae  conforming  to  Test  2  (i.e.  V  (1)  X  (2) 
is  201  and  V(ll)  X  (12)  is  211). 

It  may  be  shown  that,  in  any  such  quartet,  the  crosses 
of  the  two  pairs  of  time  antitheses  are  factor  antitheses 
of  each  other  and  the  crosses  of  the  two  pairs  of  factor 
antitheses  are  time  antitheses  of  each  other. 

We  are  now  ready  to  follow  through  the  complete  or 
double  rectification  of  all  formulae.  This  is  obtained 
by  crossing  the  crosses  and  gives  the  same  result  in  which- 
ever order  it  is  done,  —  whether  first  crossing  the  time 
antitheses  and  then  crossing  the  results,  or  first  crossing 
the  factor  antitheses  and  then  crossing  the  results,  and 
the  result  is  the  same  as  the  fourth  root  of  the  product 

1  Algebraically,  the  proof  is  given  in  Appendix  I  (Note  B  to  Chapter 
VII,  §  8). 


146         THE  MAKING  OF  INDEX  NUMBERS 


Rectified  Arithmetic  and  Harmonic,  Simple     / 


(Prices) 


13  H  75  76  IT  70 

CHART  26P.  The  upper  tier  are  the  curves  of  a  quartet  of  related  for- 
mulae. The  next  tier  are  formed  by  welding,  or  crossing  geometrically, 
each  pair  of  time  antitheses  (1  and  11  yielding  101;  2  and  12  yielding 
102) ;  the  next,  by  welding  each  pair  of  factor  antitheses  (1  and  2  yielding 
201;  11  and  12  yielding  211);  and  the  last,  by  welding  all  four  in  the 
upper  tier  (or  both  in  the  second  or  both  in  the  third).  No  one  of  the 
upper  tier  fulfills  either  test;  those  of  the  second  fulfill  Test  1  but  not  Test 
2;  those  of  the  third  fulfill  Test  2  but  not  Test  1 ;  those  of  the  last  ful- 
fill both  tests. 


RECTIFYING  FORMULA  BY  "  CROSSING"    147 

of  the  entire  quartet.1  Thus  this  fourth  root,  the  double 
rectification  of  any  of  the  quartet  of  formulae,  must  satisfy 
both  tests. 

Rectified  Arithmetic  and  Hamcnic.  Simple 

(Quantities) 


73  */4  75  '16  '17 

CHART  26Q.     Analogous  to  Chart  26P. 


'id 


A  doubly  rectified  formula  is  numbered  300,  increased 
by  the  number  of  the  lowest  numbered  of  the  quartet 
of  formulae  from  which  it  is  derived.  All  the  relation- 
ships may  be  illustrated  by  the  following  scheme  for  the 
quartet  Formulae  1,  11,  2,  12,  above  cited. 

1  See  Appendix  I  (Note  to  Chapter  VII,  §  9). 


148 


THE  MAKING  OF  INDEX  NUMBERS 


1 

crossed  with 

11 

gives         101 

crossed 
with 

crossed 
with 

crossed 
with 

2 

crossed  with 

12 

gives         102 

gives 
201 

crossed  with 

gives 
211 

gives 
gives         301 

§  10.  Numerical  Illustration 

We  may  illustrate  all  three  rectifications  by  taking  the 
figures  for  1917  for  the  quartet  of  Formulae  1,  11,  2,  12. 
These  are  for  the  index  numbers  of  prices : 


(1)  =  175.79 

(2)  =  152.75 


(11)  =  157.88 

(12)  =  172.11 


The  geometric  means  or  rectifications  of  the  time  an- 
titheses are 


(101)  =  VI  X  11  =  V175.79  X  157.88  =  166.60 

(102)  =  V2  X  12  =  V152.75  X  172.11  =  162.14 

It  is  interesting  to  observe  that  these  results  conform- 
ing to  Test  1  are  not  so  far  apart  as  the  original  figures 
which  do  not  so  conform. 

Similarly  the  rectifications  respecting  Test  2  are 

(201)  =      Vr~X~2  =  V175.79  X  152.75  =  163.87 
(211)  =  Vll  X  12  =  V157.88  X  172.11  =  164.84 

It  is  interesting  to  observe  that  these  figures  conform- 
ing to  Test  2  are  closer  together  than  the  original  figures 
which  do  not  so  conform. 


RECTIFYING  FORMULAE  BY  "CROSSING"    149 

Finally,  the  complete  rectification  gives 

(301)    =  VlOl  X  102  =  V166.60  X  162.14  =  164.35 

=  V201  X  211   =  V163.87  X  164.84  =  164.35 

=  V/1X2X11X12  = 

V  175.79  X  152.75  X  157.88  X  172.11  =  164.35 

§  11.   Graphic  Illustration 

Charts  26P  and  26Q  give  the  rectification  of  the  simple 
arithmetic  and  harmonic,  i.e.  of  Formulae  1,  11,  2,  12  (in 
which  quartet  1  and  11  are  time  antitheses  of  each  other, 
as  are  2  and  12,  while  1  and  2  are  factor  antitheses  of 
each  other,  as  are  11  and  12).  These  four  are  drawn 
from  the  same  origin  (upper  part  of  the  figure),  the  factor 
antitheses,  or  even  numbered,  being  dotted  lines. 

Their  rectifications  by  Test  1  are  drawn  immediately 
below,  the  dark  curve  101  being  the  rectification  of  For- 
mulae 1  and  11 ;  and  the  dotted  curve  102  being  the  rec- 
tification of  2  and  12.  These  two  rectified  formulae  agree 
with  each  other  better  than  the  original  formulae. 

The  third  tier,  on  the  other  hand,  gives  the  rectifica- 
tions by  Test  2,  201  being  the  rectification  of  Formulae 
1  and  2,  and  211  of  11  and  12.  These  two  also  are  closer 
than  the  first  four. 

Finally,  the  lowest  tier  gives  301,  the  completely  rectified 
index  number.  It  may  be  considered  as  the  rectification 
by  Test  2  of  the  pair  rectified  by  Test  1,  or  it  may  be 
considered  as  the  rectification  by  Test  1  of  the  pair  recti- 
fied by  Test  2,  or  it  may  be  considered  as  the  rectifica- 
tion of  the  whole  original  quartet  by  both  tests  at  once. 

Thus  each  rectification  splits  a  difference  and  each 
index  number  represented  by  the  final  curve  is  the  geo- 
metric average  of  the  four  from  which  it  is  derived.  These 
methods  of  rectification  by  crossing  apply  generally. 


150 


THE  MAKING  OF  INDEX  NUMBERS 


§  12.  Rectifying  Simple  Geometric,  Median,  Mode,  and 
Aggregative  by  Both  Tests 

Graphically,   Charts    27P   and   27Q   show   the   recti- 
fication of  the    simple    geometric.    This   is    a   shorter 


Rectified  Geometric.  Simple 

(Prices) 


73  74  75  Y*  77  13 

CHART  27P.  Analogous  to  Chart  26P ;  but  the  quartet  21,  21,  22,  22 
contains  two  duplicates,  so  that  the  upper  tier  of  four  curves  reduce  to 
two;  the  two  of  the  second  tier  simply  repeat  those  last  named  and  the 
two  curves  in  the  third  tier  reduce  to  one.  The  one  in  the  lower  tier  merely 
repeats  the  last  named. 

process  than  that  shown  hi  the  last  section  as  the  sim- 
ple geometric  already  meets  Test  1  and  only  needs  recti- 
fication by  Test  2.  But,  for  uniformity,  we  put  in  all  four 
steps,  the  first  " rectification"  being,  in  this  case,  merely 
a  repetition  of  the  formulae  ;  for  we  may  regard  Formula 
21  as  its  own  time  antithesis,  and  22  as  its  own. 


RECTIFYING  FORMULA  BY  "CROSSING"    151 

That  is,  the  first  tier  gives  the  quartet, 

21  21 

22  22 

(Formula  21  being  the  time  antithesis  of  21  and  Formula 
22  of  22,  while  one  of  the  22's  is  the  factor  antithesis  of 
one  of  the  21's  and  the  other  22  of  the  other  21).  In 

Rectified  Geometric.  Simple 

(Quantities) 


yj  74  vs  ve  77  ^ 

CHART  27Q.    Analogous  to  Chart  27P. 

the  second  tier,  Formula  121  is  the  " horizontal"  recti- 
fication of  Formulae  21  and  21,  i.e.  is  identical  with  21, 
and  likewise,  Formula  122  is  the  "horizontal"  rectifica- 
tion of  Formulae  22  and  22,  i.e.  is  identical  with  22.  The 
third  tier,  221,  is  supposed  to  represent  two  coincident 
formulae,  one  the  "vertical"  rectification  of  one  pair, 
Formulae  21  and  22,  and  the  other  of  the  other  pair,  21 
and  22.  The  fourth  tier  is  evidently  identical  with  the 
third,  being  the  rectification  of  Formulae  221  and  221  (as 
well  as  of  121  and  122). 

Were  it  not  for  the  fact  that  usually  we  have  four 
really  distinct  formulae  to  rectify  we  would  omit  two  of 


152 


THE  MAKING  OF  INDEX  NUMBERS 


these  tiers  (the  second  and  last) ;  for  the  only  real  recti- 
fication is  by  Test  2. 

Charts  28P  and  28Q  show  in  exactly  the  same  way 
the  rectification  of  the  simple  median,  and  Charts  29P 
and  29Q  that  of  the  simple  mode,  and  Charts  30P  and 
30Q  that  of  the  simple  aggregative. 


Rectified   Median.  Simple 

(Prices) 


'13  74  V5  V*  '//  V* 

CHART  28P.     Analogous  to  Chart  27P  as  to  duplications. 

§  13.    Results  of  Doubly  Rectifying  Simples 

Graphically,  Charts  31P  and  31 Q  show  at  a  glance 
the  rectification  of  all  simples  (modes  omitted).  The 
top  tier  simples  are  only  1  and  11  because  the  mode  (41) 
is  omitted;  and  because  21,  31,  51,  already  conforming 
to  Test  1,  are  postponed  to  the  second  tier,  where 
they  occur  as  121,  131,  151,  along  with  those  rectified 
by  Test  1.  All  rectified  by  Test  2  are  in  the  third  tier; 


RECTIFYING  FORMULAE  BY  "CROSSING"    153 

while  the  last  gives  those  rectified  by  both  tests.  It 
will  be  seen  that  Curves  301  and  321  are  practically  paral- 
lel everywhere  except  1917-1918,  where  Curve  301  (fixed 
base)  still  bears  evidence  of  the  original  distortion  due 
to  one  commodity,  skins.  These  two  are  fairly  similar  to 
331,  while  351  stands  alone.  Formula  341  (omitted)  has 
comparatively  little  resemblance  to  the  rest. 

Rectified  Median.  Simple 

(Quant  Hies) 


73  74  75  7$  17  '18 

CHART  28Q.    Analogous  to  Chart  28 P. 

Thus  we  may  say,  in  general,  that  by  rectifying  simple 
index  numbers  we  secure  a  moderate,  but  only  a  moderate, 
degree  of  agreement  among  the  three  principal  formulae. 
That  this  agreement  is  not  better  is  because,  the  simples 
involve  such  outlandish  weighting  that  they  are  almost 
incorrigible.  This  is  especially  true  of  the  aggregative 
Formula  51  with  its  "haphazard"  weighting,  which  has  no 
relation  to  the  weighting  employed  by  the  others. 

Moreover,  the  rectification  of  the  simples  by  Test  2 


154 


THE  MAKING  OF  INDEX  NUMBERS 


involves  a  practical  absurdity.  Simple  index  numbers 
of  prices  have  an  excuse  for  existing  only  when  we  have 
no  knowledge  of  what  weights  could  be  used,  that  is,  no 


Rectified  Mode.  Simple 

(Prices) 


\5% 


73  74  75  '16  '17  75 

CHART  29P.    Analogous  to  Chart  27P  as  to  duplications. 

knowledge  of  the  "q's"  and  so  no  knowledge  of  the  values, 
p0qo,  etc.  But  rectifying  a  simple  index  number  of 
prices  by  Test  2,  on  the  other  hand,  requires  its  factor 
antithesis  obtained  by  dividing  the  corresponding  simple 


RECTIFYING  FORMULAE  BY  "  CROSSING"    155 

index  number  of  quantities  into  the  value  ratio.  This 
implies  that  we  do  know  the  quantities  and  values.  But 
if  we  had  all  this  knowledge  we  would,  in  practice,  use 
it  at  the  start,  and  employ  a  better  system  of  weighting 
than  the  simple  weighting. 

Rectified  Mode.  Simple 

(Quantities) 


73 


75 

CHART  29Q. 


7$  '17 

Analogous  to  Chart  29P. 


Nevertheless,  for  completeness,  I  have  included  in 
this  book  the  rectification  of  simples.  It  serves  to  show 
how,  even  starting  with  the  handicap  of  absurd  weight- 
ing, we  can  achieve  a  very  considerable  rectification,  though 
we  can  never  completely  overcome  the  handicap. 


156 


THE  MAKING  OF  INDEX  NUMBERS 


§  14.  Rectifying  the  Weighted  Arithmetic  and 
Harmonic  by  Both  Tests 

Far  more  important,  therefore,  are  the  rectifications 
of  the  weighted  index  numbers. 

The  consideration  of  the  first  two  quartets  on  the  list, 
consisting  of  Formulae  3,  19,  4,  20,  and  5,  17,  6,  18,  is 


Rectified    Aggregative.  Simple 
(Prices) 


13  '14  '15  '/*  17  18 

CHART  SOP.    Analogous  to  Chart  27P  as  to  duplications. 

postponed.    The  reason  is  that,  in  each  case,  their  recti- 
fication is  identical  with  that  of  Formulae  53  and  54. 

Graphically,  Charts  32P  and  32Q  show  the  rectification 
of  the  arithmetic-harmonic  quartet,  Formulae  9,  13,  10, 
14.  By  Test  1  in  Charts  32P  and  32Q  we  roll  or  weld 
Curves  9  and  13  into  109,  and  10  and  14  into  110.  Again, 
by  Test  2  Curves  9  and  10  are  compressed  into  209,  and  13 
and  14  into  213.  Finally,  by  putting  all  four  through 
both  rolling  mills  (in  either  order,  or  both  at  once),  we 


RECTIFYING  FORMULA  BY  "CROSSING"    157 

roll  them  together  into  the  single  curve  309  at  the  bottom 
of  the  diagram. 

Charts  33P  and  33Q  show  the  same  process  by  which 
the  quartet,  Formulae  7,  15,  8,  16,  are  passed  through 
our  rolling  mills  to  be  welded  into  the  fully  rectified  307. 

Rectified    Aggregative.  Single 

(Quantities) 


74  '/$ 

CHART  30Q. 


77 


Analogous  to  Chart  SOP. 

Chart  33  resembles    Chart   32   very   closely   in   every 
detail. 

§  15.    Rectifying  the  Weighted  Geometric,  Median, 
Mode,  and  Aggregative  by  Both  Tests 

Graphically,  Charts  34P  and  34Q  show  the  rectified 
quartet  of  geometries,  Formulae  23,  29,  24,  30.  These 
charts  resemble  Charts  32  and  33  except  that  the  four 
formulae  to  start  with  are  only  about  half  as  far  apart. 

Charts  35P  and  35Q  show  the  rectification  quartet  of 
the  geometric  Formulae  25,  27,  26,  28,  and  resemble 
closely  Chart  34  in  every  detail. 

Charts   36P,  36Q   and   37P,  37Q  give   the   rectified 


158         THE  MAKING  OF  INDEX  NUMBERS 


Simple  Index  Number*  of  Prices 
and    Their  Antitheses   and  Derivatives. 


Satisfying  neither  test. 
test  I  o/f/y. 

••  ? 

both   tests 
(modes  omitted) 


'14 


76 


77 


CHART  3 IP.  This  double  rectification  of  all  the  simple  index  numbers 
(of  which  21,  31,  51  and  their  factor  antitheses  22,  32,  52  are  omitted  in 
the  first  tier,  being  inserted  in  the  second  tier  as  121,  131,  151  and  122, 
132, 152)  results  in  only  a  moderate  degree  of  agreement,  as  the  lowest  tier 
of  curves  indicates. 


RECTIFYING  FORMULA  BY   " CROSSING"    159 


Simple  Index  Numbers  of  Quantities 

ond  Their  Antitheses  and  Derivatives. 

Satisfying  neither  test, 
test  I  on/y. 


\5% 


'14  75  75  77 

CHART  31Q.    Analogous  to  Chart  31P. 


160 


THE  MAKING  OF  INDEX  NUMBERS 


medians  as  indicated.  They  are  like  the  figures  for  the 
preceding  except  that  they  are  usually  still  closer  to  each 
other  than  the  geometric  but  less  consistently  so. 

Rectified  Arithmetic  and  tiarmenic,  Weighted 


(By 


Values 
(Prices) 


One  Year) 


13  74  75  16  17  Id 

CHART  32P.  A  quartet  of  widely  differing  weighted  index  numbers, 
with  scattered  chain  figures,  combined  by  rectification  into  309,  which  is 
practically  identical  with  the  other  rectified  index  numbers  that  follow,  and 
the  chain  figures  of  which  practically  coincide  with  the  fixed  base  figures. 

Charts  38P  and  38Q  give  the  rectified  mode  for  the 
quartet  of  Formulae  43,  49,  44,  50  as  well  as  of  45,  47,  46, 
48 ;  for  these  two  do  not  need  separate  charts,  being  iden- 
tical up  to  the  limit  of  our  calculations.  That  is,  43  is 
practically  identical  with  45,  44  with  46,  49  with  47,  and 
50  with  48.  Thus  the  mode  does  not  respond  appreciably 
to  changing  weights. 


RECTIFYING  FORMULAE  BY  "  CROSSING "    161 

Charts  39P  and  39Q  show  the  rectification  of  the  two 
weighted  aggregatives,  the  formulae  of  Laspeyres'  (53) 
and  Paasche's  (54)  which  recur  again  and  again  in  our 
system  of  formulae.  It  may  be  considered  as  the  recti- 
fication, not  only  of  the  quartet,  Formulae  53,  59,  54,  60, 

Rectified  Arithmetic  and  Harmonic.  Weighted 

(By  Values  in  One  Year) 
(Quantitlts) 


\S% 


'&  74  75  7*  77 

CHART  32Q.    Analogous  to  Chart  32P. 


78 


but  also  of  the  quartets,  3,  19,  4,  20,  and  5,  17,  6,  18,  all 
three  quartets  being  identical. 

This  rectification,  like  some  of  the  preceding,  is  not 
really  of  four  but  of  two  only.  Also,  unlike  the  case  of 
the  others,  there  is  only  one  real  rectification ;  that  is, 
the  first  rectification  and  second  are  identical  with  each 
other  as  well  as,  of  course,  with  the  two  together.  Hence 
there  is  only  the  one  identical  curve  for  each  of  the  three 
lower  tiers. 


162         THE  MAKING  OF  INDEX  NUMBERS 

§  16.  Results  of  Double  Rectifications  of  Weighted 
Index  Numbers 

Graphically,  Charts  40P  and  40Q  show  at  a  glance  the 
rectification  of  all  the  weighted  index  numbers  (modes 

Rectified  Arithmetic  and  Harmonic,  Weighfed 

(By  "Mixed"  Value*) 
(Prices) 


•&  '/*  'IS  VS  V7  Vd 

CHAET  33P.    Analogous  to  Chart  32P  except  that  the  weighting  is  by 
mixed  or  "hybrid"  values. 

omitted).  The  agreement  thus  brought  about  among 
the  weighted  index  numbers  is  far  greater  than  that 
brought  about  among  the  simples.  In  fact,  all  these 
rectified  weighted  index  numbers  agree  perfectly  for 
practical  purposes.  If  the  medians  were  excluded  the 
eye  could  scarcely  detect  any  discordance.  Only  the 


RECTIFYING  FORMULAE  BY  " CROSSING"    163 

rectified  weighted  modes   (omitted  from  chart)   really 
disagree  with  the  rest. 

Charts  41 P  and  41 Q  outline  the  limits  of  the  various 
weighted  formulae  (omitting  modes  and  medians),  showing 
that  the  limits  contract  as  the  tests  are  fulfilled.  This 
diagram  shows  that  all  weighted  index  numbers  (omitting 

Rectified  Arithmetic  and  Harmonic,  Weighted 

(By  "M/xec/*  Values  ) 
(Quantities) 


75  74  'IS  16  '17  73 

CHAKT  33Q.    Analogous  to  Chart  33P. 

modes  and  medians)  lie  within  limits  far  closer  together  than 
the  original  price  relatives  or  quantity  relatives  averaged. 
What  is  more  important,  it  shows  that  this  range  is 
greatly  reduced  when  at  least  one  of  the  two  tests  is  met. 
Finally,  it  shows  that  those  which  satisfy  both  tests 
lie  within  an  amazingly  small  range,  so  small  as,  for  prac- 
tical purposes,  to  be  entirely  negligible. 
Charts  42P  and  42Q  give  individually  the  doubly  recti- 


164 


THE  MAKING  OF  INDEX  NUMBERS 


fied  weighted  index  numbers  (modes  omitted).  It  will 
be  noted  that  the  eye  can  scarcely  detect  any  lack  of 
parallelism,  except  slightly  in  the  case  of  the  median. 


Rectified  Geometric,  Weighted 

(dy  Values  in  One  Year) 
(Prices) 


75  74  75  7$  77  16 

CHAHT  34P.    Analogous  to  Chart  32P  except  that  all  of  the  quartet  are 
of  geometric  derivation  instead  of  arithmetic  and  harmonic. 


§  17.  List  of  Quartets 

Let  us  now  again  "take  account  of  stock,"  and  list, 
first,  all  the  quartets,  and  then  all  the  formulae.  The 
following  is  a  complete  list,  omitting  duplicates,  of  all 
the  quartets  which  may  be  formed  from  the  46  primary 
formulae  by  matching  each  formula  with  its  antitheses. 


RECTIFYING  FORMULAE  BY  "CROSSING"    165 

Arithmetic  and  Harmonic 


1 

2 

11 
12 

7 
8 

15 
16 

9 
10 

13 
14 

giving  Formula  301 
giving  Formula  307 
giving  Formula  309 


Rectified  Geometric.  Weighted 

(By  Values  In  One  Year) 
(Quantities) 


}•* 


13 


'14  75  16  77 

CHART  34Q.    Analogous  to  Chart  34P. 


Geometric 


21 
22 

21 
22 

23 
24 

29 
30 

25 

26 

27 
28 

giving  Formula  321 
giving  Formula  323 
giving  Formula  325 


166 


THE  MAKING  OF  INDEX  NUMBERS 
Median 

giving  Formula  331 
giving  Formula  333 
giving  Formula  335 


31 
32 

31 
32 

33 
34 

39 
40 

35 
36 

37 
38 

Rectified    Geometric,  Weighted 

(By 'Mixed*  Values) 
(Prices) 


73  74  75  7«  77  '18 

CHABT  35P.    Analogous  to  Chart  ,34P  except  that  the  weighting  is  by 
mixed  or  "hybrid"  values. 


Mode 


41 
42 

41 
42 

43 
44 

49 
50 

45 
46 

47 

48 

giving  Formula  341 
giving  Formula  343 
giving  Formula  345 


RECTIFYING  FORMULAE  BY  "CROSSING"     167 


Aggregative 


51 
52 

51 
52 

53 
54 

59 
60 

giving  Formula  351 
giving  Formula  353 


Rectified    Geometric.  Weighted 

(By  m  Mixed"  Va/i/es) 

(Quantities) 


13  '14  t$  IB  17 

CHART  35Q.    Analogous  to  Chart  35P. 


The  omitted  duplicates  are 


and 


19 
20 


17 
18 


168 


THE  MAKING  OF  INDEX  NUMBERS 


Rectified  Median.  Weighted 
(By  Values  in  One  Year) 
(Prices) 


73  74  75  7*  77 

CHABT  36P.    Analogous  to  Chart  32P. 

which  are  identical  with 


53 
54 


59 
60 


all  of  which  identical  quartets  merely  contain  Laspeyres* 
and  Paasche's  formulae  in  their  various  r61es.  Each  of 
the  last  three  might  be  written 


L 
P 


P 
L 


RECTIFYING  FORMULAE  BY  "  CROSSING "    169 


All  these  identical  quartets  remind  us  again  that 
Laspeyres'  and  Paasche's  formulae  are  time  antitheses  of 
each  other  and  also  factor  antitheses  of  each  other,  as 
well  as  that  they  are  arithmetic,  harmonic,  and  aggre- 
gative. 


Rectified  Median.  Weighted 

(By  Values  in  O/*  Year) 

(Quantities) 


^  34 


75  74  75  '16  ;<7> 

CHABT  36Q.    Analogous  to  Chart  36P. 


'Iff 


Of  the  quartets  it  has  doubtless  been  observed  by  the 
reader  that  some  really  reduce  to  duets,  namely,  those 
quartets  resulting  in  Formulae  321,  331,  341,  351  (in  which 
cases  the  two  numbers  in  the  same  horizontal  line  are  iden- 
tical) ;  and  also  the  quartet  resulting  in  Formula  353 
(in  which  case  the  diagonals  are  identical,  being  Las- 
peyres' and  Paasche's  formulae).  Formula  353, 


170 


THE  MAKING  OF  INDEX  NUMBERS 


which  is  identical  with  the  12  formulae  indicated  in  §  7, 
will  hereafter  be  referred  to  only  as  353. 


Rectified    Median.  Weighted 

(By  "Mixed"  Value*) 
(Prices) 


73 


74 


75 


'IB 


CHART  37P.  Analogous  to  Chart  36P  except  that  the  weighting  is  by 
mixed  or  "hybrid"  values. 

§  18.  List  of  Formulae  thus  far  Obtained 

The  complete  list  of  formulae,  including  the  primary, 
those  fulfilling  Test  1,  those  fulfilling  Test  2,  and  those 
fulfilling  both  tests,  are  given  in  Table  12,  in  which 
duplications  are  omitted  (being  indicated  only  by  a  dash) . 

In  this  table  any  formula  (like  21)  which  already  fulfills 


RECTIFYING   FORMULAE  BY  "CROSSING"    171 

Test  1  before  crossing  is  pushed  forward  and  appears 
later  (as  121) ;  and  likewise  any  (like  221)  which  fulfills 
both  tests  after  only  one  kind  of  crossing  is  pushed  for- 
ward and  appears  later  (as  321).  That  is,  in  this  table 
the  numbers  with  "300"  comprise  those  and  those  only 

Rectified  Median.  Weighted 

(By  'Mixed"  Values) 
(Quantities) 


355 


73  74  V5  '16  77  W 

CHABT  37Q.    Analogous  to  Chart  37P. 

which  fulfill  both  tests;  the  numbers  with  "200"  com- 
prise those,  and  those  only,  which  fulfill  only  Test  2 ;  the 
numbers  with  "100"  comprise  those,  and  those  only, 
which  fulfill  only  Test  1,  while  the  numbers  less  than  100 
include  those,  and  those  only,  which  fulfill  neither  test. 

Thus  far  we  have  assembled  for  examination  the  follow- 
ing number  of  formulae : 

46  primary  formulse,  including  eight  (21,  22,  31,  32,  41, 
42,  51,  52)  which  conform  to  Test  1  and  so  are  pushed 


172         THE  MAKING  OF  INDEX  NUMBERS 
Rectified  Mode.  Weighted 


(Pricts) 


44.50  (or  46.46) 
43, 4$  (or  45.47) 


243,  24$(or  24S,2*?I 


343  (or  345) 


19  74  75  7*  '17  Y« 

CHART  38P.    Analogous  to  Charts  32P  and  33 P. 

forward  (to  121,  122,  131,  132,  141,  142,  151,  152)  in 
the  last  table  of  formulae ; 

19  new  derivative  formulae  (derived  by  crossing  time 
antitheses  among  the  primary)  conforming  to  Test  1 
and  including  one  (153)  which  conforms  to  Test  2  as 
well  and  so  is  pushed  forward  (to  353)  in  the  last  table ; 

22  new  derivative  formulae  (derived  by  crossing  factor 
antitheses  among  the  primary)  conf orming  to  Test  2 ; 
9  new  derivative  formulae  conforming  to  both  tests. 


RECTIFYING  FORMULAE  BY  "CROSSING"    173 

This  makes  96  separate  formulae,  of  which  38  conform 
to  neither  test,  26  conform  to  Test  1  only,  18  conform  to 
Test  2  only,  and  14  conform  to  both  tests.  This  list  of 
96  formulae  constitutes  our  main  series  of  formulae  and 

Rectified  Mode.  Weighted 
(Quantities) 


44.50  for  46.409 
43,49  (or45,47t 


243.249(or245l247t 


5% 


'13  74  '8  1*  '17  78 

CHART  38Q.    Analogous  to  Chart  38P. 

includes  most  of  the  important  kinds.  Certain  other 
formulae  which  will  be  considered  later  are,  in  each  case, 
closely  similar  to  some  of  these  96  varieties. 

In  a  later  chapter  all  these  and  other  forms  of  index 
numbers  will  be  systematically  compared.  But  already 
one  important  conclusion  forces  itself  upon  us.  It  is 


174 


THE  MAKING  OF  INDEX  NUMBERS 


one  which  has  already  been  noted,  namely,  that,  after 
rectification,  the  great  discrepancies  which  we  first  noticed 
among  index  numbers  constructed  by  different  formulae 
tend  to  disappear;  and  that  excepting  the  modes  and  the 
index  numbers  derived  from  simples,  all  the  index  numbers 
thus  farfoundwhich  obey both  tests  agree  closely  with  each  other. 

Rectified  Aggregative.  Weighted 

(Prices) 


75 


76 


77 


73 


CHAET  39P.  Analogous  to  Chart  32P,  but  the  quartet  53,  59,  54,  and 
60  contains  two  duplicates;  consequently  the  four  curves  in  the  upper 
tier  reduce  to  two ;  the  two  of  the  second  tier  reduce  to  one ;  and  the  two 
lower  tiers  merely  repeat  the  preceding. 


§  19.  Other  Methods  of  Crossing 

In  this  chapter  the  " cross"  between  any  two  formulae 
has  always  been  the  geometric  mean  between  those  two 
formulae.  And  we  have  seen  that  this  geometric  mean 
satisfied  the  test  in  question.  That  is,  the  geometric 
mean  of  two  time  antitheses  satisfies  the  time  test,  and 
the  geometric  mean  of  two  factor  antitheses  satisfies  the 
factor  test.  If  we  try  the  arithmetic  mean  or  the  harmonic 


RECTIFYING  FORMULAE  BY  "CROSSING"    175 


mean  of  the  two  antitheses,  it  will  fail  to  satisfy  the  re- 
quired test. 

Algebraically,  this  is  readily  proven  by  applying  the 
usual  twofold  routine  by  which  we  have  tested  any  for- 
mula. 

Take,  for  example,  Formulae  53  and  54  or  59.  If  we  cross 
these  two  formulae  arithmetically  instead  of  crossing  them 


Rectified  Aggregative.  Weighted 

(Quantities) 


53.60  (-53) 


\s% 


/J  14  15  16  17 

CHART  39Q.    Analogous  to  Chart  39P. 

geometrically,  we  obtain 

o  I 


Starting  with  this  formula,  let  us  apply  to  it  Test  1,  by 
means  of  the  usual  twofold  procedure  : 

Interchanging  the  "O's"  and  the  "1's," 

^  + 


Inverting, 


176         THE  MAKING  OF  INDEX  NUMBERS 


Weighted  Index  Number*  cf  Prices 

and  Their  Antitheses   and  Derivatives 


neither  t*st 
test   I  on/if 


both    testy 
(modes  omitted) 


7.J  7*  75  W  17  73 

CHART  40P.  Analogous  to  Chart  3 IP,  except  that  the  double  rectifi- 
cation of  these  weighted  index  numbers  results  in  a  much  closer  agreement 
than  was  the  case  with  the  simples. 

The  resulting  formula  is  not  the  original  arithmetic  but 
the  harmonic.  Therefore,  the  original  formula  fails  to 
conform  to  Test  1,  and  the  resulting  (harmonic)  formula 
is  its  time  antithesis. 

The  reader  can  readily  prove  that  the  same  formula 
also  fails  to  satisfy  Test  2.  In  this  case  the  twofold  pro- 
cedure consists,  as  we  know,  in  interchanging  the  "p's" 


RECTIFYING  FORMULA  BY   "CROSSING"     177 
Weighted  Index  Numbers  of  Quantities 

and  Their  Antitheses  ond  Derivatives. 


Satisfying  neither  test, 
test   I  on/y. 

both  tests, 
(motes  omitted) 


«j  y*  •&  if  17  '& 

CHART  40Q.    Analogous  to  Chart  40P. 

and  the  "q's"  and  dividing  the  result  into  the  value 
ratio    P1*?1  .    The  formula  resulting  from  this  twofold  pro- 


cedure  for  Test  2  will,  it  may  surprise  the  reader  to  find, 
be,  in  this  case,  the  same  as  the  above  formula  resulting 
from  the  twofold  procedure  for  Test  1.  That  is,  the  orig- 
inal Fand  final  formulae  (arithmetic  and  harmonic  crosses 
of  53  and  54)  are  not  only  tune  antitheses  of  each  other 
but  also  factor  antitheses  of  each  other. 

In  passing,  we  may  note  that  these  two  formulae,  the  re- 
sulting and  the  original,  are  listed  in  the  Appendix  as  For- 
mulae 8053  and  8054,  being  factor  antitheses  of  each  other. 

If  we  should  test  the  harmonic  crossing  we  would  sim- 
ply reverse  the  above  process.  We  would  start  with  8054 
and  reach  8053.1 

1  See  Appendix  I  (Note  A  to  Chapter  VII,  §  19). 


178 


THE  MAKING  OF  INDEX  NUMBERS 


Range  of  Prices 

and  of 

Three  Types  of  Index  Numbers 


Weighted,  Satisfymtj  neither  test. 
•  •»  on/i/  /  or  only  2. 

»  -         both  /  and  f 

(tnodfj   and  med/am  ora> OtO <) 


14 


16 


77 


18 


CHART  41  P.  The  limits  of  the  weighted  index  numbers  contract  markedly 
as  the  tests  are  fulfilled. 

All  of  the  above  examples  are  of  the  aggregative  type. 
What  we  have  found  is  that  if  two  aggregatives  are 
crossed  arithmetically  (or  harmonically)  the  resulting 
cross  will  not  satisfy  either  Test  1  or  Test  2. 

By  like  testing  of  the  other  types  of  index  numbers,  — 


RECTIFYING  FORMULA  BY   "  CROSSING  " 
Range  of  Quantities 

and  of 

TTiree  lypes  of  Index  Numbers 

Weighted,  Satisfying   ntithtr  test. 
~  "         on/i/  /  or  only  2. 

toth  /  and  2. 
(modes  ana  medians  omitted) 


179 


l*ft  *•* 


73          14          75          76  77          73 

CHABT  41Q.    Analogous  to  Chart  41P. 


180         THE  MAKING  OF  INDEX  NUMBERS 


Weighted  Index  Numbers  Doubly  Rectified 
(Modes  Omitted) 

(Prices) 


CHAET  42P.    This  chart  shows  separately  the  seven  resultant  curves 
(lowest  tier)  of  Chart  40P,  with  the  chain  figures  added. 


arithmetic,  harmonic,  geometric,  median,  and  mode,  — 
we  find  that  crossing  arithmetically  or  harmonically  any 
two  time  antitheses  (i.e.  Formulae  3  and  19,  5  and  17,  4  and 
20,  6  and  18,  13  and  9,  15  and  7,  14  and  10,  16  and  8, 
23  and  29,  25  and  27,  24  and  30,  26  and  28,  33  and  39, 
35  and  37,  34  and  40,  36  and  38,  43  and  49,  45  and  47, 
44  and  50,  46  and  48)  will  yield  formulae  which  likewise 
fail  to  satisfy  either  test. 

We  can  thus  convince  ourselves  that  not  a  single  one 
of  the  46  primary  index  numbers  can  be  arithmetically 


RECTIFYING  FORMULA  BY  " CROSSING"    181 

Weighted  Index  Numbers  Doubly  Rectified 
(Modes  Omitted) 

(Quantities)  _355 


7J  %?  VS  "16  77 

CHART  42Q.    Analogous  to  Chart  42P. 


(or  harmonically)  crossed  with  its  antithesis  (whether 
time  or  factor)  and  yield  a  result  which  will  satisfy  either 
test.  The  only  question  remaining  is,  may  any  among 
them  be  successfully  crossed  by  any  other  method  than 
geometrically  ? 

We  need  scarcely  consider  any  other  methods  of  cross- 
ing index  numbers  than  the  six  types  of  averages  which 
we  have  considered  for  index  numbers  themselves.  Of 
these  six  we  have  already  considered  three.  The  remain- 
ing three  are  the  median,  mode,  and  aggregative. 

As  to  using  the  median  or  modal  method  for  crossing 
two  formulae,  obviously  this  is  impossible.  No  such 
averages  exist  when,  as  in  the  present  problem,  only  two 
terms  are  to  be  averaged. 

There  remains  the  aggregative  method  of  crossing 
two  index  numbers.  This  method  is  inapplicable  as  a 


182         THE  MAKING  OF  INDEX  NUMBERS 


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RECTIFYING  FORMULAE  BY  " CROSSING"    183 

method  of  averaging  Formulae  3  and  19,  or  any  of  the  other 
pairs  of  antitheses,  except  the  geometric  and  aggregative 
index  numbers ;  because,  except  in  these  two  cases,  there 
are  no  appropriate  numerators  and  denominators  of  the 
terms  to  be  averaged  such  as  are  required  to  fit  into  an 
aggregative  formula. 

The  reader  who  is  interested  will  find  these  two  cases 
discussed  in  the  Appendix.1 

§  20.   Historical 

Except  in  the  case  of  Formula  353,  the  history  of  which 
will  be  especially  noted  later,  no  crosses  of  formulae,  such 
as  those  set  forth  in  this  chapter,  seem  to  have  been  pre- 
viously pointed  out. 

Instead  of  crossing  the  formulae  themselves,  previous 
students  of  index  numbers  have  crossed  their  weights,  as 
will  be  shown  in  the  next  chapter. 

1  See  Appendix  I  (Note  B  to  Chapter  VII,  §  19). 


CHAPTER  VIII 
RECTIFYING  FORMULA  BY  CROSSING  THEIR  WEIGHTS 

§  1.  Introduction 

THE  foregoing  list  of  96  formulae  thus  far  obtained,  and 
ending  with  Formula  353,  constitutes  a  complete  system 
of  formulae,  primary  and  derivative,  which  I  shall  call 
the  "main  series."  The  additions  to  it  in  this  chapter 
are,  in  essence,  only  slight  variations  of  this  main  series. 
These  additions  are  included  in  deference  to  the  wishes 
of  other  students  of  index  numbers,  and  in  order  that  the 
list  shall  cover  all  formulae  previously  suggested  by  others 
and  all  points  of  view.  They  may  be  called  the  "sup- 
plementary series. " 

Each  of  these  additional  formulae  is  weighted  and  each 
weight  is  a  cross  between  two  other  weights.  This  cross- 
ing of  two  weights  is  merely  an  alternative  method  of 
combining  two  kinds  of  weighted  index  numbers.  To 
illustrate,  if  we  start  with  the  two  formulae,  23  and  29, 
namely  the  geometric  index  numbers,  —  one  weighted 
according  to  the  values  in  the  base  year  and  the  other 
weighted  according  to  the  values  in  the  given  year,  and 
which  are  time  antitheses  of  each  other  —  we  can  combine 
these  two  formulae  in  either  of  two  ways.  One  way  is 
that  already  described  in  the  main  series,  and  con- 
sists simply  in  crossing  the  two  index  numbers  themselves, 
i.e.  multiplying  them  together  and  extracting  the  square 
root.  The  result  is  Formula  123  of  the  main  series.  The 
other  way,  about  to  be  discussed,  is  to  construct  a  new 
formula  on  the  same  model  as  23  and  29,  such  that  each 

184 


RECTIFYING  BY  CROSSING  WEIGHTS        185 

individual  weight  is  a  cross  between  corresponding  weights 
in  23  and  29.  This  resulting  formula  is  called  1123  and 
gives,  as  we  shall  see,  virtually  the  same  result  as  123. 
The  result  of  the  first  kind  of  crossing,  such  as  For- 
mula 123,  may  be  called  a  cross  formula ;  and  that  of  the 
second,  such  as  1123,  a  cross  weight  formula. 

Numerically,  Formula  23  (for  prices  for  1917  rel- 
atively to  1913)  gives  154.08,  while  29  gives  170.44. 
Their  cross  by  the  geometric  mean,  as  per  Formula  123, 
is  V154.08  X  170.44,  or  162.05.  So  much  for  Formula 
123,  the  cross  formula  between  23  and  29. 

The  cross  weight  formula  involves  more  detail,  for 
we  must  first  cross  each  of  the  36  pairs  of  weights.  For 
bacon,  the  weight  under  Formula  23,  that  is,  the  value  of 
bacon  in  the  base  year,  1913,  is  133.117  while  the  weight 
under  29,  that  is,  its  value  in  the  given  year,  1917,  is  282.743. 
The  cross  of  these  weights  (by  the  geometric  mean)  is 
V133.117  X  282.743  =  193.86,  which  is  the  weight  we 
were  seeking  for  bacon.  Similarly,  the  weight  for  barley 
is  the  cross  between  111.607  and  276.549,  which  is  175.68 ; 
similarly,  the  weight  sought  for  beef  is  1097.04,  and  so  on. 
Next  we  calculate  a  new  index  number  based  on  these 
36  new  weights  but  otherwise  precisely  analogous  to  For- 
mulae 23  and  29.  The  result  is  found  to  be  161.62.  This 
is  by  Formula  1123. 

Algebraically,  Formula  123,  the  cross  formula,  is 
V23  X  29.  (The  reader  who  chooses  can  substitute  the 
algebraic  expressions  for  Formulae  23  and  29  as  given 
in  Appendix  V.)  On  the  other  hand,  Formula  1123  — 
the  cross  weight  formula  —  is  itself  given  fully  in 
Appendix  V.  The  reader  will  observe  that  it  is  exactly 
analogous  to  Formulae  23  and  29,  the  only  difference  being, 
that,  instead  of  the  weights  pQqQ,  p'oq'o,  etc.,  as  per  For- 
mula 23,  or  instead  of  the  weights  p\q\,  p'i0/'i,  etc.,  as  per 


186 


THE  MAKING  OF  INDEX  NUMBERS 


Formula    29,    we    now    have    the    weights 
/'nT/itf'i,  etc. 


§  2.  The  Cross  Weight  Geometries,  Medians,  and  Modes 

We  have  taken  Formula  1123  as  the  first  illustration  of  a  cross  weight 
formula.  It  was  derived  by  crossing  the  weights  in  Formulae  23  and  29 
and,  on  their  model,  writing  a  new  formula.  The  same  method  may  be 
used  for  combining  any  two  formulas  of  the  same  model  differing  only  in 
their  weights.  But,  it  is  interesting  to  observe,  if  we  thus  combine  For- 
mulas 25  and  27  we  get  identically  the  same  result  as  we  have  just  obtained 
by  combining  23  and  29;  for  the  cross  weights  in  the  first  case  are 
^(Poqo)  X  (p\qi),  etc.,  and  in  the  second,  ^(p0qi)  X  (ptfo),  etc.,  which 
are  evidently  the  same.  Thus  Formula  1123  may  be  just  as  truly  said  to 
come  from  25  and  27  as  from  23  and  29.  On  the  other  hand,  the  cross 
formula,  123,  is  made  only  from  23  and  29 ;  that  from  Formulas  25  and  27 
is  125,  which  is  slightly  different. 

Likewise  we  designate  by  1133  the  formula  derived  by  crossing  the 
weights  of  the  medians,  33  and  39,  or  of  33  and  37;  and  by  1143  that  by 
crossing  the  weights  of  the  modes,  43  and  49,  or  45  and  47.  Formula 
1133  agrees  closely  with  133  and  1143  with  143. 

The  preceding  formulas,  i.e.  the  cross  weight  geometries,  medians,  and 
modes  have  been  given  first  because  they  resemble  each  other  so  closely 
and  are  the  simplest  of  the  six  types. 

Table  13  contains  the  identification  numbers  for  the  geometries,  medi- 
ans, and  modes,  (1)  of  primary  formulas,  and  (2)  and  (3)  of  the  two  kinds 
of  derivatives  from  them  —  the  cross  formulas  and  the  cross  weight 
formulas. 


TABLE   13.     DERIVATION   OF  CROSS   FORMULAE  AND 
CROSS   WEIGHT   FORMULAE 


TYPE 

(1) 
PRIMARY 

FORMUL/E 

TO  BE  COMBINED 

COMBINED 
(2)                                (3) 

By  Crossing 
the  Two  Formula 

By  Crossing 
Their  Weights 

Geometric  .  . 

23  and  29 
25  and  27 

33  and  39 
35  and  37 

43  and  49 
45  and  47 

123 

125 

133 
135 

143 
145 

1123 
1133 
1143 

Mode    

RECTIFYING  BY  CROSSING  WEIGHTS        187 

§  3.  The  Cross  Weight  Aggregatives 

The  process  of  deriving  a  price  index  by  crossing  the 
weights  of  the  two  weighted  aggregatives  (which  we  may 
here  refer  to  as  Formulae  53  and  59)  is  slightly  different, 
since  the  weights  are  not  values  (like  p0q0  and  piqi),  but 
only  quantities  (like  qQ  and  q\).  The  resulting  formula 
is  1153  of  the  same  model  as  53  and  59,  but  with  weights 
(Vq0qi,  etc.)  which  are  the  crosses  of  their  weights.  It 
agrees  closely  with  Formula  153. 

We  have  now  considered  the  cross  weight  geometries, 
medians,  modes,  and  aggregatives. 

There  remain  only  the  arithmetics  and  harmonics, 
which  will  be  considered  shortly. 

§  4.   Comparisons  of  the  Cross  Weight  Formulae 
thus  far  Obtained 

All  the  cross  weight  types  just  given  satisfy  Test  1. 
This  may  readily  be  proved  in  the  usual  manner  by  inter- 
changing the  "0V  and  the  "IV  in  the  formulae  of  Ap- 
pendix V.  Furthermore,  each  cross  weight  formula 
agrees  almost  exactly  with  the  corresponding  cross  for- 
mula, except  the  median.  That  is,  Formula  1123  is  vir- 
tually the  same  as  123  or  125,  1143  as  143  or  145,  1153 
as  153  (  =  353).  Table  14  shows  some  of  these  similarities. 

Graphically,  the  curves  representing  cross  formulae 
and  the  curves  representing  cross  weight  formulae  are 
indistinguishable,  except  in  the  case  of  the  median,  as 
shown  in  Charts  43P  and  43Q. 

§  5.  Cross  Weight  Arithmetics  and  Harmonics 

There  remain  to  be  described  the  cross  weight  arithmetic  and  harmonic 
formulae.  These  are  numbered  1003  and  1013.  In  the  above  table 
they  are  not  represented,  as  there  were  no  corresponding  cross  for- 
mulae in  our  previous  tables,  for  the  reason,  of  course,  that  arithmetic  for- 


188         THE  MAKING  OF  INDEX  NUMBERS 


mulae  are  crossed  not  with  other  arithmetics,  but  with  harmonics,   and, 
vice  versa,  harmonics  with  arithmetics. 

Thus  Formula  103  was  a  cross  between  3  and,  not  9,  but  19 ;  Formula 
104  was  a  cross  between  4  and  20;  Formula  107  was  a  cross  between  7 
and  15 ;  etc.  But,  while  we  can  thus  cross  two  formidce,  one  of  which  is 
arithmetic  and  the  other  harmonic,  crossing  weights  of  two  formulae  im- 
plies that  they  are  both  of  the  same  model,  differing  only  in  their  weights. 
If  the  models  of  two  formulae  differ  we  would  not  know  which  model  to 

Close  Agreement  of 

Cross  Formulae  and  Cross  Weight  Formulae 
(Prices) 


'13  '14  15  '16  17  18 

CHART  43  P.  The  pairs  indicated  practically  coincide  (1103  with  103, 
1104  with  104,  etc.)  except  in  the  case  of  the  medians.  All  the  16  formulae, 
shown  in  pairs  in  these  eight  separate  diagrams,  obey  Test  1  but  not  Test  2. 

use  in  building  the  proposed  cross  weight  formula.  Thus,  if  we  should 
cross  the  weights  of  an  arithmetic  and  a  geometric  formula  we  would  not 
know  what  to  do  with  the  weights  after  we  had  them.  It  is  equally  mean- 
ingless to  cross  the  weights  of  an  arithmetic  and  harmonic. 

In  short,  crossing  weights  is  meaningless  except  as  applied  to  two  of  a 
kind,  such  as  to  two  arithmetics  or  to  two  harmonics  —  not  one  of  each ; 
and  when  it  is  applied  to  two  arithmetics  or  to  two  harmonics  the  result- 
ing cross  weight  formulae  (unlike  the  other  four  types  of  cross  weight  for- 
mulae considered  hitherto)  will  fail  to  satisfy  Test  1.  This  is  another 
interesting  result  of  the  one-sideduess  of  the  arithmetic  and  of  the  har- 
monic* 


RECTIFYING  BY  CROSSING  WEIGHTS        189 


Close  Agreement  of 

Cross-Formulae  and  Cross-Weight  Formulae 
(Quantities) 


yj  M  75  75  17 

CHART  43Q.    Analogous  to  Chart  43P. 


78 


TABLE  14.  INDEX  NUMBERS  BY  CROSS  WEIGHT  FORMULAE 
(1123,  1133,  1143,  1153)  COMPARED  WITH  INDEX  NUMBERS 
BY  CORRESPONDING  CROSS  FORMULAE  (123,  133,  143,  153) * 


PBICES 


FORMULA  No. 

1913 

1914 

1915 

1916 

1917 

1918 

123 

100. 

100.12 

99.94 

113.83 

162.05 

177.80 

1123 

100. 

100.14 

99.89 

114.17 

161.62 

177.87 

133 

100. 

100.54 

99.68 

108.12 

159.93 

173.57 

1133 

100. 

100.52 

99.57 

108.39 

162.63 

170.85 

143 

100. 

101. 

100. 

108. 

164. 

168. 

1143 

100. 

101. 

100. 

108. 

164. 

168. 

153 

100. 

100.12 

99.89 

114.21 

161.56 

177.65 

1153 

100. 

100.13 

99.89 

114.20 

161.70 

177.83 

1  Omitting,  for  brevity,  the  cross  formulae  125  and  145,  which  agree  closely  with  123 
and  143  respectively ;  and  135,  which  also  agrees  closely  with  133  except  hi  the  years  1917 
and  1918  when  the  former  is  162.00  and  178.44  as  contrasted  with  the  159.93  and  173.57 
for  133  as  given  in  the  table. 


190 


THE  MAKING  OF  INDEX  NUMBERS 


QUANTITIES 

FORMULA.  No. 

1913 

1914 

1915 

1916 

1917 

1918 

123 

100. 

99.30 

109.14 

118.92 

118.85 

125,01 

1123 

100. 

99.34 

109.07 

118.79 

118.82 

125.31 

133 

100. 

98.60 

105.58 

115.82 

118.16 

122.94 

1133 

100. 

98.71 

105.46 

115.50 

118.23 

122.27 

143 

100. 

97. 

103. 

103. 

98. 

124. 

1143 

100. 

97. 

103. 

103. 

98. 

124. 

153 

100. 

99.33 

109.10 

118.85 

118.98 

125.37 

1153 

100. 

99.33 

109.08 

118.82 

118.86 

125.29 

Thus,  Formulse  1003,  1004,  in  which  cross  weights  are  used  to  unite 
pairs  of  arithmetics,  and  Formulae  1013,  1014,  in  which  they  are  likewise 
applied  to  harmonics,  correspond  to  no  cross  formulae  given  in  our  main 
series.  This  is  why  we  have  numbered  them  1003,  etc.,  and  not  1103, 
etc.  If  we  wish  to  construct  cross  formulas  which  correspond  to  the  new 
cross  weight  1003,  1004,  1013,  1014,  we  need  to  cross  3  and  9;  4  and  10; 
13  and  19;  14  and  20.  This  is  done  in  Table  15  for  the  purpose  of 
comparison. 


TABLE  15.  INDEX  NUMBERS  BY  CROSS  WEIGHT  FORMULAE 
(1003,  1004,  1013,  1014)  COMPARED  WITH  CORRESPONDING 
CROSS  FORMULAE 

(1913  =  100) 


FORMULA  No. 

PRICES 

1914 

1915 

1916 

1917 

1918 

^3  X9 

1003 

100.43 
100.45 

100.99 
100.93 

116.17 
116.02 

171.14 
170.81 

182.46 
182.54 

>/4  X  10 
1004 

99.51 
99.47 

98.52 
98.60 

112.71 

112.84 

157.98 
158.01 

173.30 
173.03 

Vl3  X  19 
1013 

99.79 
99.81 

98.96 
98.91 

112.67 
112.53 

153.96 
153.51 

172.95 
173.02 

Vl4  X  20 
1014 

100.87 
100.83 

101.03 
101.10 

115.43 
115.54 

165.19 
165.24 

183.74 
182.94 

Here,   as  before,   the  cross  weight  formulae  and   the   cross   formulae 


RECTIFYING  BY  CROSSING  WEIGHTS        191 

agree  almost  perfectly.  They  represent,  essentially,  two  different  routes 
toward  the  same  result.  Neither  satisfies  Test  1  (nor  Test  2  for  that 
matter). 


§  6.  The  Cross  Weight  Formulae  Derived  from  the 
Factor  Antitheses  of  the  Preceding 

In  our  weight  crossing,  in  the  case  of  the  geometries,  medians,  and  modes, 
we  have  taken  only  the  odd  numbered  formulae.  But  we  may,  in  like 
manner,  cross  the  weights  employed  in  Formulae  24  and  30  (i.e.  in  their 
denominators)  and  so  build  up  a  new  formula  on  their  model.  This  is 
called  Formula  1124.  It  is  also  the  cross  weight  formula  from  Formulae 
26  and  28.  Likewise  we  derive  Formula  1134  from  34  and  40  (or  from 
36  and  38),  and  Formula  1144  from  44  and  50  (or  from  46  and  48). 

We  have  now  derived,  as  our  complete  list  of  cross  weight  formulae  of 
odd  numbers:  Formulae  1003,  1013,  1123,  1133,  1143,  1153  and  also  those 
to  which  we  have  given  the  corresponding  even  numbers  :  Formulae  1004, 
1014,  1124,  1134,  1144,  1154.  But  all  except  Formula  1154  of  the  latter 
six  even  numbered  formulae  were  derived,  not  as  antitheses  (although 
such  they  are)  l  of  the  six  corresponding  odd  numbered  formulae.  They 
were  derived  directly  by  crossing  the  weights  of  4  and  10,  14  and  20,  24 
and  30,  34  and  40,  44  and  50. 


§  7.  Cross  Weight  Arithmetics  and  Harmonics  are 
not  Truly  Rectified 

As  stated,  the  arithmetic  Formula  1003  is  not  analogous  to  the  truly 
rectified  103;  nor  is  1013.  There  is  no  way  whatever,  through  weight 
crossing  alone,  to  rectify  the  arithmetics  alone  or  the  harmonics  alone 
relatively  to  Test  1.  To  get  truly  rectified  formulae  the  above  results 
(1003  and  1013)  have  still  to  be  crossed  with  each  other.  That  is,  in  this 
case,  the  method  of  crossing  weights  must  be  eked  out  by  the  method  of 
crossing  formulae.  Crossing,  then,  1003  and  1013,  we  obtain  a  new  for- 
mula, numbered  1103,  which  does  satisfy  Test  1  and  is  the  nearest  ap- 
proach to  a  cross  weight  formula  analogous  to  the  cross  formula,  103. 
Moreover,  their  results  practically  coincide.  Similarly  (by  crossing  1004 
and  1014),  we  get  the  new  Formula  1104,  corresponding  to  104. 

Having  reached  Formulae  1103  and  1104,  we  insert  them  in  Chart  43 
as  the  nearest  analogues  of  103  and  104.  We  note  that,  here  again,  the 
results  of  rectifying  by  crossing  weights  on  the  one  hand,  and  by  crossing 
the  formulaB  themselves  on  the  other,  coincide  to  all  intents  and  purposes. 

We  may  now  add  to  Table  13  in  §  2  the  following ; 


See  Appendix  I  (Note  to  Chapter  VIII,  §  6). 


192         THE  MAKING  OF  INDEX  NUMBERS 


PRIMARY 

COMBINED 

TTPB 

TO  BE 

COMBINED 

By  Crossing  the 
Two  Formulae 

By  Crossing  Their 
Weights 

By  Crossing  the 
Two  Formulae  in 
the  Last  Column 

Arithmetic  

3  and  9 

omitted  »  \ 

1003 

1 

5  and  7 
13  and  19 

omitted1  J 
omitted  l  \ 

1013 

>    1103 

Arithmetic  and  Harmonic 

15  and  17 

3  and  19 
5  and  17 

omitted  *  J 

103 
105 

impossible 

1  "Omitted"  means  that  no  identification  number  was  given  to  these  crosses  as  they 
serve  no  purpose  in  our  main  series.  But  the  figures  for  some  of  these  formulae  (for  prices, 
fixed  base)  were  calculated  and  given  in  §  5  above. 


Close  Agreement  of  Cross-formulae  and 

Cf of s- Weight  formulae  (Fully  Rectified) 
{Prices) 


"/J  M  15  16  17  IB 

CHABT  44P.    Analogous  to  Chart  43P,  except  that  here  both  tests  are 
fulfilled. 


§  8.  List  of  the  Formulae  Obeying  Test  1  Derived 
Partly  or  Wholly  by  Weight  Crossing 

We  see  that  the  arithmetic  1003  and  the  harmonic  1013,  and  their 
factor  antitheses,  1004  and  1014,  all  derived  by  weight  crossing,  had 
to  be  used  merely  as  a  preliminary  scaffolding  for  building  1103,  derived 
partly  by  formula  crossing.  After  discarding  the  scaffolding  our  new 
formulae  are  1103,  1123,  1133,  1143,  1153,  and  these  supplementary  for- 


RECTIFYING  BY  CROSSING  WEIGHTS        193 

mulae  agree  almost  precisely  with  their  mates  (103,  123,  133,  143,  153)  in 
the  main  series.  Their  factor  antitheses  (the  next  even  numbered  for- 
mulae, 1104,  1124,  1134,  1144,  1154)  likewise  agree  closely  with  their  mates 
(104,  124,  134,  144,  154). 

§  9.  Rectifying  the  New  Formula  by  Test  2 

The  new  formulae,  1103,  1123,  1133,  1143,  1153  (and  their  factor  an- 
titheses, the  next  even  numbered  formulae)  all  satisfy  Test  1,  as  do  the 
corresponding  formulae  in  the  main  series.  But  not  a  single  one  of  them 
satisfies  Test  2  (although  in  the  main  series  one  formula,  the  analogue 
of  1153,  namely,  153,  does  satisfy  Test  2). 

Close  Agreement  of  Cross-Formulae 

Crasx-Weight  Formulae  (Fully  Rectified) 
(Quantities) 


VJ  '14  75  75  '17  Id 

CHART  44Q.    Analogous  to  Chart  44P. 

In  order  to  obtain  conformity  to  Test  2  we  must  further  rectify  and, 
for  this  purpose,  the  only  process  of  combining  the  factor  antitheses  is  by 
crossing  the  formulae  themselves.  Crossing  their  weights  is  inapplicable 
because  the  two  formulae  to  be  combined  are,  in  every  instance,  of  different 
models.  The  doubly  rectified  formulae  numbers  are  given  in  the  last 
column  of  Table  16. 

These  pairs  of  corresponding  formulae  satisfying  both  tests  agree  with 
each  other  even  more  perfectly  than  did  the  pairs  satisfying  only  Test  1 
agree  with  each  other.  That  is,  Formula  303  agrees  almost  exactly  with 
1303,  Formulae  323  and  325  with  1323,  etc. 

Graphically,  Charts  44  show  the  almost  perfect  iden- 
tity of  1303  with  303  and  of  1323  with  323,  of  1333  with 
333,  1353  with  353,  and  1343  with  343  (or  would,  were 
the  last  two  indicated). 

Moreover,  we  may  note  in  passing,  that  the  entire  group 


194 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  16.     DOUBLY  RECTIFIED  FORMULAE  DERIVED  FROM 
PRIMARY  WEIGHTED  FORMULAE 


WHOLLY  BY  FORMULA 
CROSSING 

RESULTS  IN  MAIN 
SERIES 

PARTLY  BY  WEIGHT 
CROSSING 

RESULTS   IN 
SUPPLEMENTARY 
SERIES 

•tnq    _  ir)4    _  -me          -iric 

—  qno  —  one 

(  =  153  =  154) 
^107  X  1U8 

(=353) 
=  307 

V1103  X  1104 

=  1303 

Vi09  X  110 

=  309 

N/123  X  124 

000    \ 

Vl25  X  126 
Vl33  X  134 

=  325  / 
=  333\ 

^1123  X  1124 

=  1323 

^135  x  136 
Vi43  X  144 

=  335J 
=  343  1 

^1133  X  1134 

=  1333 

Vi45  X  146 
153  =  154 

=  345  / 
=  353 

^1143  X  1144 

=  1343 

(  -  103  =  104  =  105  =  106) 

(  »  303  =  305) 

V1153  X  1154 

=  1353 

of  rectified  formulae,  by  both  methods  of  crossing,  agrees 
almost  absolutely,  excepting  only  those  originating  from 
modes  and  medians,  and  even  the  medians  agree  with  the 
rest  well  enough  for  most  practical  purposes.  This  re- 
markable agreement  is  clear  from  a  study  of  the  figures 
for  all  the  index  numbers  given  in  Appendix  VII  and  will 
be  emphasized  later. 

§  10.  Several  Methods  for  Crossing  Weights  as  Con- 
trasted with  Only  One  (in  General)  for  Crossing  the 
Formulae  Themselves 

At  the  close  of  the  last  chapter  it  was  shown  that  to 
cross  formulae  the  geometric  method  of  crossing  is  uni- 
versally appropriate  (although  in  two  instances  the  aggre- 
gative method  would  also  be  applicable).  But  in  weight 
crossing  the  geometric  method  has  no  such  preeminence 
for  we  can  equally  well,  in  all  cases,  use  the  arithmetic 
method  or  the  harmonic  method  without  prejudicing 
the  conformity  of  the  result  to  the  test  which  we  are 


RECTIFYING  BY  CROSSING  WEIGHTS        195 

seeking  to  meet.  In  this  chapter,  I  have  employed  the 
geometric  method  of  crossing  weights  chiefly  because  it 
is  the  form  of  crossing  hitherto  most  in  favor.  The  other 
methods  are  discussed  in  the  Appendix.1  They  include 
several  interesting  and  ingenious  suggestions  which  writers 
on  index  numbers  have  made.  But  only  one  of  them  has 
much  practical  value.  That  one  (2153)  is  useful  as  a 
short  cut  approximation  to  353. 

§  11.   Conclusions 

In  this  chapter  we  have  obtained  the  following  new 
formulae:  1003,  1004,1013,1014,  1103,  1104,  1123,  1124, 
1133,  1134,  1143,  1144, 1153, 1154, 1303,  1323, 1333,  1343, 
1353 ;  and,  in  the  Appendix,  2153,  2154,  2353,  3153,  3154, 
3353,  4153,  4154,  4353.  Of  these,  all  coincide  approxi- 
mately with  the  middle  tine  of  our  fork,  excepting  the  arith- 
metics, Formulae  1003,  1004  (which  have  an  upward 
and  downward  bias  respectively  and  which  fall  on  the 
mid-upper  and  mid-lower  tines) ;  and  also  excepting  the 
harmonics,  1013,  1014  (which  have  a  downward  and  up- 
ward bias  respectively  and  which  fall  on  the  mid-lower  and 
mid-upper  tines);  and  also  excepting  the  modes,  1143, 
1144,  1343,  which  are  erratic;  and,  excepting  also, 
possibly,  the  medians,  1133,  1134,  1333,  which  ar^ 
slightly  erratic. 

From  what  has  been  said  it  is  now  clear  that  crossing 
the  weights  of  two  formulae  of  the  same  model  and  so 
forming  a  new  formula  of  that  same  model  yields  almost 
identically  the  same  numerical  result  as  crossing  the  for- 
mulas themselves.  It  is  also  clear  that  formula  crossing 
is  a  process  which  can  be  applied  to  any  two  formulae 
whether  the  two  be  of  the  same  model  or  not,  whereas 
weight  crossing  cannot  be  used  except  where  the  two 

*  See  Appendix  I  (Note  to  Chapter  VIII,  §  10). 


196         THE  MAKING  OF  INDEX  NUMBERS 

formulae  to  be  combined  are  built  on  exactly  the  same 
model,  differing  only  in  their  weights, 
i  In  other  words,  formula  crossing  is  a  universal  method 
of  compromising  between  two  formulae,  while  weight 
crossing  is  of  restricted  application.  We  found  it  incapable, 
for  instance,  of  rectifying  any  formula  by  Test  2,  and 
even  incapable  of  rectifying  some  formulae  by  Test  1.  In 
short,  weight  crossing  is  never  necessary  and  is  some- 
times inapplicable. 

§  12.  Historical 

It  is  rather  odd,  therefore,  that  hitherto  the  simpler 
and  more  universally  serviceable  of  the  two  processes 
has  been  almost  wholly  overlooked.  The  reason  is  his- 
torical tradition.  In  the  history  of  the  index  numbers 
the  first  stage  was  to  discuss  the  virtues  of  the  simple 
index  numbers,  chiefly  the  arithmetic  and  geometric. 
The  next  step  was  to  assign  weights  supposed  to  be  rep- 
resentative of  the  conditions  prevailing  in  the  periods 
concerned.  Drobisch  was,  apparently,  the  first  to  make 
use  specifically  of  the  quantities  of  two  years  compared 
by  an  index  number. 

Following  this  line  of  study,  Scrope  and  Walsh  pro- 
posed the  cross  weight  aggregative  formula  here  num- 
bered 1153,  and  Walsh  also  1154;  Marshall  and  Edge- 
worth  proposed  the  cross  weight  aggregative,  Formula 
2153;  Walsh,  Formula  2154;  Lehr,  Formulae  4153  and 
4154,  and  Walsh  the  cross  weight  geometric,  Formula 
1123.  Of  the  cross  formulae,  8053  (see  Appendix  V)  was 
suggested  by  Drobisch  and  Sidgwick.  Finally,  Formula 
353,  of  which  more  will  be  said  later,  was  first  mentioned, 
though  not  at  that  time  advocated,  by  Walsh. 


CHAPTER  IX 
THE  ENLARGED  SERIES  OF  FORMULAS 

§  1.  Introduction 

THUS  far  we  have  accomplished  three  chief  things. 
We  have  shown : 

(1)  That  there  are  two  important  reversibility  tests 
of  index  numbers ;    ' 

(2)  That  certain  formulae  have  a  "  bias  "  or  constant 
tendency  to  err  relatively  to  Test  1 ; 

(3)  That  any  formula  whatever  can  be  "  rectified  "  so 
as  to  conform  to  either  test  or  both. 

In  the  course  of  this  study,  we  have  constantly  added 
to  the  number  of  formulas  demanding  consideration. 
Before  proceeding  to  compare  all  these  formulas  as  to  their 
relative  accuracy,  we  may  now  pause  to  "  take  account  of 
stock  "  and  also  complete  our  list  by  the  addition  of  ten 
more  formulas. 

We  first  set  forth  the  main  series  of  96  formulas  (original 
and  derivative)  of  which  those  having  identification 
numbers  between  1  and  99  were  the  primary  formulas; 
those  having  identification  numbers  between  100  and  199 
conformed  to  Test  1 ;  those  having  identification  numbers 
between  200  and  299  conformed  to  Test  2;  and  those 
having  identification  numbers  between  300  and  399  con- 
formed to  both  Test  1  and  Test  2.  The  last  and  culminat- 
ing one  of  these  formulas,  96  in  number,  was  Formula  353, 


We  shall  find  this  to  be  theoretically  the  best  formula.1 

1  For  model  examples  to  aid  in  the  practical  calculation  of  this  as  well 
as  eight  other  sorts  of  index  numbers,  see  Appendix  VI,  §  2. 

197 


198         THE  MAKING  OF  INDEX  NUMBERS 

To  this  list  of  96  formulae  we  have  just  added  a  supple- 
mentary list  of  28  more  formulae,  which  owe  their  origin 
to  the  process  called  weight  crossing  (in  place  of  formula 
crossing  employed  in  the  main  series). 

These  28  new  formulae  are  as  follows :  Those  numbered  between  1000 
and  1999,  originated  in  crossing  weights  geometrically  two  by  two ;  those 
between  2000  and  2999,  originated  in  crossing  them  arithmetically;  those 
between  3000  and  3999,  originated  in  crossing  them  harmonically;  those 
between  4000  and  4999,  originated  in  crossing  them  by  means  of  a  special 
weighted  arithmetical  average. 

To  these  124  formulae  we  now  add  ten  miscellaneous 
formulae,  which  make  38  in  the  supplementary  series, 
in  addition  to  the  96  in  the  main  series,  or  134  in  all. 

They  are :  Those  between  5000  and  5999,  formed  by  crossing  formulae 
in  the  "300"  list ;  those  between  6000  and  6999,  formed  by  using  a  broader 
base  than  one  year ;  those  between  7000  and  7999,  formed  by  averaging  the 
six  forms  of  Formula  353  obtained  by  using  each  of  the  six  years  as  base ; 
those  between  8000  and  8999,  formed  by  crossing  formulae  arithmetically 
and  harmonically ;  those  between  9000  and  9999,  formed  by  using  round 
numbers  as  weights. 

More  specifically,  these  final  ten  miscellaneous  formulas  are  as  follows : 
As  to  the  5000's : 

Formula  5307  is  the  cross  between  Formulae  307  and  309 ; 

Formula  5323  is  the  cross  between  Formulae  323  and  325 ; 

Formula  5333  is  the  cross  between  Formulae  333  and  335 ; 

Formula  5343  is  the  cross  between  Formulae  343  and  345. 
As  to  the  6000's  : 

Formulae  6023  and  6053  are  like  23  and  53  respectively,  except  that 
instead  of  the  first  year  being  the  base,  the  base  is  an  average  made  up  of 
two  or  more  years. 

Formula  7053  is  an  average  of  six  forms  of  353,  with  six  different  bases. 

Formula  8053  is  the  arithmetic  average  of  53  and  54.  Formula  8054 
is  the  harmonic  average  of  the  same,  as  well  as  the  factor  antithesis  of 
8053.  It  may  be  shown  that  the  cross  between  Formulae  8053  and  8054  is 
identical  with  353. * 

We  may  classify  the  134  formulae  which  have  been 
noted.  They  will  be  classified  under  five  heads,  according 
as  they  owe  their  origin  to  (1)  arithmetics  and  harmonics, 
(2)  geometries,  (3)  medians,  (4)  modes,  (5)  aggrega- 
tives. 

1  See  Appendix  I  (Note  to  Chapter  IX,  §  1). 


THE  ENLARGED   SERIES  OF  FORMULAE       199 


§  2.  List  of  the  Arithmetic  and  Harmonic  Formulae 

The  first  group,  to  which  Table  17  is  devoted,  includes  the  two  types, 
the  arithmetics  and  the  harmonics,  since  in  the  crossing,  which  we  found 
necessary,  these  two  could  not  be  kept  apart. 

The  two  upper  lines  relate  to  the  simples  and  their  derivatives,  and  the 
eight  lines  following  relate  to  the  weighted  and  their  derivatives.  The 
first  column  gives  the  arithmetic,  the  second,  the  harmonic,  the 
third,  the  derived  cross  formula  satisfying  Test  1,  the  fourth  and  fifth, 
the  cross  formulae  satisfying  Test  2,  the  sixth,  the  cross  formulae  satis- 
fying both  tests,  thus  completing  the  arithmetic  and  harmonic  formulae 
in  the  main  series.  The  remaining  columns  give  the  cross  weight  formula  and 
their  crosses. 

A  dash  indicates  a  formula  omitted  because  duplicated  elsewhere. 
These  duplications  are  given  below  Table  17.  In  the  same  way,  the  du- 
plications of  Tables  18,  19,  20,  and  21  are  given  below  them. 


TABLE   17.     ENLARGED  ARITHMETIC-HARMONIC   GROUP 


PRIMARY 
FORMULAE 

CROSS  FORMULAE 

CROSS  WEIGHT  FORMUI^E  AND  THEIR 
CROSSES 

^ 

•s|  . 

i 

In 

•*» 

Q 

•s-s 

4 

a 

£ 

By  Test  2 

1 

Arith. 

Harm. 

I  -5  * 

"S 

1 

5 

H 

£ 

ft 

ol 

^ 
H 

lis 

1 

11 

101 

201 

211 

301 

2 

12 

102 

— 

13 

— 

— 

213 

— 

1003 

1013 

1103 

1303 

— 

14 

— 

1004 

1014 

1104 

— 

15 

— 

.  — 

215 

— 

— 

16 

— 

7 

— 

107 

207 

— 

307 

5307 

8 

— 

108 

9 

— 

109 

209 

— 

309 

10 

— 

110 

Duplications  (indicated  above  by  dashes)  : 

3  =  53  17  =  53  103  =  353  203  -  353  303  =  353 

4  =  54  18  =  54  104  =  353  205  =  353  305  =  353 

5  =  54  19  =  54  105  =  353  217  =  353 

6  =  53  20  =  53  106  -  353  219  =  353 


The  above  table  covers  two  of  our  six  types.     Each  of  the  following 
four  tables  covers  one.     The  next  three  are  alike  in  form. 


200 


THE  MAKING  OF  INDEX  NUMBERS 


|  3.  List  of  the  Geometric,  Median,  and  Mode 
Groups  of  Formulae 

The  following  three  tables  give  lists  of  all  the  formulae  in  the  geometric, 
median,  and  mode  groups,  including  all  derivatives. 

TABLE   18.    ENLARGED  GEOMETRIC  GROUP 


PRIMARY 
FORMULA 

CROSS  FORMULAE 

CROSS  WEIGHT  FORMULA  AND 
THEIR  CROSSES 

CROSS  OF 

323  AND 

325 

By  Test  1 

By  Test  2 

By  Both 

Cross  Weight 
Formulae 

Their  Cross 

__ 

121 

_ 

321 

— 

122 

23 

123 

223 

323 

1123 

1323 

5323 

24 

124 

1124 

25 

125 

225 

325 

26 

126 

27 

227 

28 

29 

229 

30 

Duplications  (indicated  above  by  dashes)  : 

21  =  121     221  -  321 

22  =  122 

TABLE  19.    ENLARGED  MEDIAN  GROUP 


PRIMARY 
FORMULAE 

CROSS  FORMULA 

CROSS  WEIGHT  FORMULA 
AND  THEIR  CROSSES 

CROSS  OP 

L333  AND 

335 

By  Test  1 

By  Test  2 

By  Both 

Cross 
Weight 
Formulae 

Their  Cross 

^_ 

131 

__ 

331 

— 

132 

33 

133 

233 

333 

1133 

1333 

5333 

34 

134 

1134 

35 

135 

235 

335 

36 

136 

37 

237 

38 

39 

239 

40 

Duplications  (indicated  above  by  dashes)  : 

31  =  131    231  -  331 

32  =  132 


THE  ENLARGED  SERIES  OF  FORMULAE      201 


TABLE  20.    ENLARGED  MODE  GROUP 


PRIMARY 
FORMULAE 

CROSS  FORMTJLuE 

CROSS  WEIGHT  FORMULA 
AND  THEIR  CROSSES 

CROSS  OF 

343  AND 
QJR 

Cross 

By  Test  1 

By  Test  2 

By  Both 

Weight 
Formulae 

Their  Cross 

O£O 

_ 

141 



341 

— 

142 

43 

143 

243 

343 

1143 

1343 

5343 

44 

144 

1144 

45 

145 

245 

345 

46 

146 

47 

247 

48 

49 

249 

50 

Duplications  (indicated  above  by  dashes)  : 

41  =  141     241  =  341 

42  =  142 


§  4.  List  of  the  Aggregative  Formulae 

Finally,  we  have  the  aggregative  group. 

TABLE  21.  ENLARGED  AGGREGATIVE  GROUP 


CROSS  FORMULA 

CROSS  WEIGHT  FORMTTL^E  AND  THEIR 

PRIMARY 

FORMULAE 

By  Test  1 

By  Test  2 

By  Both 

Cross  Weight 
Formulae 

Their  Cross 

__ 

151 



351 

— 

152 

53 

— 

— 

353 

1153 

1353 

54 

— 

1154 

Duplications  (indicated  above  —  except  59,  60,  259  omitted  —  by  dashes) : 

51  =  151     153  =  353     251  =  351 

52  =  152     154  =  353     253  =  353 

59  =  54  259  =  353 

60  =  53 

The  preceding  lists  do  not  include  certain  other  forms  discussed  in 
the  Appendix,1  namely,  Formula  2153,  the  cross  weight  by  the  arithmetic 
method  of  crossing ;  3153,  the  cross  weight  by  the  harmonic  method;  4153, 

1  See  Appendix  I  (Note  to  Chapter  VIII,  §  10). 


202         THE  MAKING  OF  INDEX  NUMBERS 

the  cross  weight  by  Lehr's  method  of  taking  a  weighted  arithmetic  average 
of  the  weights;  the  factor  antitheses  (2154,  3154,  4154)  of  these  three  cross 
weight  formulae  and  the  rectifications  (2353,  3353,  4353)  by  crossing  said 
antitheses  (2153  with  2154,  etc.).  Besides  these  are  a  few  other  mis- 
cellaneous forms  (6023,  6053,  7053,  8053,  8054,  9051). 

§  5.  The  Seven  Classes 

The  134  formulae  constitute  the  enlarged  series  of  for- 
mulas embracing  all  considered  in  this  book.  Including 
duplicates  the  number  is  170 ;  besides  these  there  are  the 
five  formulae  (9001, 9011,  9021, 9031,  9041)  given  in  Appen- 
dix V,  §  3. 

Our  problem  now  is  to  examine  and  discriminate  between 
these  134  formulae,  —  in  particular  to  explain  their 
differences  and  to  select  the  best.  These  134  index  num- 
bers have  been  classified  by  type,  weighting,  and  method 
of  crossing.  We  may  also,  for  convenience  in  our  dis- 
cussion, classify  them  under  the  following  seven  groups : 

S,  the  simple  index  numbers  and  their  derivatives, 

M,  the  medians  and  modes  and  their  derivatives, 

2  +,  all  other  weighted  index  numbers  having  a  double 
upward  bias, 

2  — ,  all  other  weighted  index  numbers  having  a  double 
downward  bias, 

1  +,  all  other  weighted  index  numbers  having  a  single 
upward  bias, 

1  —  ,  all  other  weighted  index  numbers  having  a  single 
downward  bias, 

0,   all  other  weighted  index  numbers  having  no  bias. 

These  seven  groups  are  mutually  exclusive,  except 
that  the  simple  modes  and  the  simple  medians,  and  then: 
derivatives,  are  included  under  both  the  first  two  headings. 

§  6.  The  Formulae  Grouped  under  the  Seven  Classes 

The  following  is  a  list  of  the  formulae  in  each  of  the  first  two  groups : 
r  Group  "S":  1,  2,  11,  12,  101,  102,  201,  211,  301,  21,  22,  321,  31,  32, 
331,  41,  42,  341,  51,  52,  351. 


THE  ENLARGED  SERIES  OF  FORMULA      203 

Closely  associated  with  the  "S"  group,  though  not  strictly  members  of 
it,  are :  9001,  9021,  9031,  9041,  9051.1 

Group  "  M  "  :  3 1-40  inclusive, 

133-136  inclusive, 
233,  235,  237,  239, 
331,  333,  335, 
1133,  1134,  1333, 
5333. 

41-50  inclusive, 
143-146  inclusive, 
243,  245,  247,  249, 
341,  343,  345, 
1143,  1144,  1343, 
5343. 
(31,  32,  331,  41,  42,  341  are  in  both  the  "S"  and  "M"  groups.) 

The  other  five  groups,  (i.e.  excluding  "S"  and  "M  ")  fall  in  the  five  tines 
of  our  five-tined  fork  (or,  if  we  wish  to  avoid,  so  far  as  possible,  any  blurring 
of  the  tines,  two  such  forks,  one  for  the  odd,  and  the  other  for  the  even 
numbered  formula),  according  to  Table  22. 

The  formulae  hold  their  approximate  positions  on  the 
"  five-tined  fork  "  wholly  according  to  the  following  fixed 
rules : 

Those  which  have  no  bias  lie  approximately  in  coinci- 
dence and  constitute  the  middle  tine.  Those  which  have 
only  one  upward  bias,  whether  type  bias  or  weight  bias, 
likewise  agree  and  form  the  mid-upper  tine.  Similarly, 
those  which  have  only  one  downward  bias,  whether  type 
or  weight,  make  the  mid-lower  tine.  Those  which  have 
a  double  upward  bias,  i.e.  a  type  bias  and  a  weight  bias, 
make  the  uppermost  tine.  Likewise  those  doubly  biased 
downward  make  the  lowermost  tine. 

The  case  of  a  downward  bias  of  one  sort  and  an  upward 
bias  of  another  being  combined  is  also  provided  for.  Such 
a  curve  turns  out  to  have  no  bias  at  all,  being  merely 
erratic.  Therefore,  it  also  lies  on  the  middle  tine.  For- 
mula 3  is  one  of  these.  As  an  arithmetic  type  it  has  an 
upward  bias,  but  having  weight  7  it  has  a  downward  bias, 

1  Given  in  Appendix  V,  §  3. 


204 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  22.    THE  FIVE-TINED  FORK 


TINE 

ARITHMETIC 

HARMONIC 

GEOMETRIC 

AGGREGATIVE 

Uppermost  (2  -f-) 

7,9 

14,16 

Mid-upper  (1  +) 

1003 

1014 

24,  26,  27,  29 

Middle         (0) 

3=6  =  (L), 
4  =  5  =  (P) 

17=20=(L), 
18  =  19=(P) 

53=60  =  (L), 
54  =  59  =  (P) 

107,  108,  109,  110, 
1103,  1104 

123,  124,  125, 
126,1123,1124 

1153,  1154 

207,  209 

213,  215 

223,  225, 
227,  229 

2153,  2154, 
3153,  3154, 
4153,  4154 

203=205  = 

217=219  = 

323,  325, 
1323 

153  =  154  = 
OCQ     ocq 

103—  104—  105—  106— 
303=305=(\/LXP) 

307,  309,  1303 

353=VLXP, 
1353, 
2353,  3353, 
4353,  5307, 
5323,  6053, 
7053,  8053, 
8054 

Mid-lower  (1  —  ) 

1004 

1013 

23,  25,  28,  30 

Lowermost  (2  —  ) 

8,  10 

13,  15 

and  the  two  neutralize ;  for,  after  cancellation,  3  reduces 
to  53,  which  is  of  such  a  type  that  we  cannot  accuse  it  of  a 
proneness  to  err  up  rather  than  down  or  down  rather  than 
up. 

Thus,  barring  "  simples  "  and  "  modes "  and  their 
derivatives  (and  possibly  medians  if  we  wish  to  have  our 
results  very  close),  we  find  that,  although  we  have  numer- 
ous formulae,  they  all  fall  under  only  five  clearly  defined 


THE  ENLARGED  SERIES  OF  FORMULA      205 

heads,  namely,  those  without  bias,  those  with  single  bias 
up  or  down,  and  those  with  double  bias  up  or  down. 

The  five  tines  include  all  the  arithmetic,  harmonic, 
geometric,  and  aggregative  weighted  index  numbers  and 
their  derivatives  which  we  have  obtained. 


CHAPTER   X 
WHAT  SIMPLE  INDEX  NUMBER  IS  BEST? 

§  1.    Introduction 

OUR  next  problem  is  to  compare  the  numerous  formulae 
which  we  have  found  and  to  select  the  theoretically  best 
formula  or  formulae,  i.e.  the  most  accurate.  This  problem 
may  conveniently  be  subdivided  into  two  parts,  viz. : 

1.  Assuming  that  we  have  no  weights  available  so  that 
we  are  compelled  to  use  simple  averages,   which  index 
number  then  is  best  ? 

2.  Assuming,  on  the  contrary,  that  we  do  have  the 
data  for  assigning  unequal  weights,  which  index  number 
then  is  the  best  ? 

In  this  chapter,  we  shall  take  up  the  first  of  these 
two  problems.  The  assumption  that  there  are  no  data 
for  weights  at  once  removes  from  our  list  of  index  numbers 
of  prices  all  the  even  numbered  ones,  and  those  derived  from 
them;  since  each  of  these  was  obtained  by  dividing  an 
index  number  of  quantities  into  a  ratio  of  values,  and, 
therefore,  presupposes  a  knowledge  of  values  and  quanti- 
ties, which  are  the  data  for  assigning  weights. 

Obviously,  also,  our  assumption  rules  out  all  the  weighted 
index  numbers  and  their  derivatives.  The  only  index 
numbers  now  left  are :  Formulae  1,  11,  21,  31,  41,  51,  101. 
Our  problem,  therefore,  reduces  itself  to  selecting  the 
best  from  these  seven  formulae. 

§  2.  Discarding  the  Two  Biased  Formulae 

Proceeding  by  a  process  of  elimination,  we  may  discard 
Formulae  1  and  11  as  they  possess  an  upward  and  a  down- 

206 


WHAT  SIMPLE  INDEX  NUMBER  IS  BEST?    207 

ward  bias  respectively.  This  has  been  proved  by  Test  1, 
the  time  reversal  test.  Formulae  21,  31,  41,  51,  and  101 
meet  successfully  Test  1,  as  has  also  been  proved.  Our 
hypothesis,  that  no  data  for  quantities  or  values  are 
available,  prevents  the  application  of  Test  2,  the  factor 
reversal  test,  since  this  involves  a  knowledge  of  values. 

§  3.   Freakishness 

So  far  as  meeting  tests  is  concerned,  therefore,  all  five 
of  the  remaining  formulae  stand  on  an  equality.  If  we  are 
to  discriminate  further,  it  must  be  on  some  other  basis. 
Such  a  basis  is  what  we  have  called  freakishness.  All 
index  numbers  may  be  assumed  to  be  somewhat  erratic, 
that  is,  no  one  is  certain  to  be  absolutely  correct.  But 
some  can  be  shown  to  be  more  erratic  than  others,  that 
is,  more  likely  to  err.  A  formula  which  can  be  shown  to 
be  especially  erratic,  as  compared  with  other  formulae,  has 
been  called  freakish. 

A  biased  formula  errs  in  a  given  direction.  An  erratic 
formula  or  a  freakish  formula  may  err  in  either  direction. 

§  4.  Discarding  Formula  51  as  Freakish 

Formula  51  may  be  discarded  as  freakish.  As  has  been 
noted  before,  while  its  weighting  is  called  simple,  it /is  not 
simple  in  the  same  sense  as  the  other  four  formulae.  In 
these  four  formulae  the  price  relatives  have  equal  weights. 
But  in  Formula  51  it  is  the  prices  themselves  which  have 
equal  weights.  Consequently,  unlike  the  other  four  index 
numbers,  Formula  51  is  affected  by  a  change  in  the  unit 
in  which  any  price  is  quoted.  Its  simple  weighting  is 
thus  quite  arbitrary,  or,  as  Walsh  says,  "  haphazard." 

As  Formula  51  is  applied  by  Bradstreet,  for  instance, 
the  unit  of  each  commodity  is  a  pound.  The  index 
number  is  found  by  taking  the  sum  of  the  prices  per  pound 


208         THE  MAKING  OF  INDEX  NUMBERS 

of  a  certain  bill  of  goods.  A  pound  of  silver  and  a  pound 
of  coal  are  counted  as  of  equal  importance.  If  the  units 
used  in  market  quotations  were  employed  so  that  the  sum 
was  made  up  of  the  price  per  ounce  of  silver  and  the  price 
per  ton  of  coal,  the  result  would  be  quite  different. 

In  the  case  of  the  aggregative,  I  doubt  even  whether  the 
general  substitution  of  the  pound  for  articles  usually 
measured  in  other  units  produces  any  improvement. 
Most  large  units,  like  the  ton  or  bale,  are  applied  to  coal 
and  hay  merely  to  lift  up  the  quotation  to  a  figure  com- 
parable to  those  in  which  the  smaller  units  are  measured. 
In  other  words,  we  avoid  quoting  hay  per  pound  because 
the  resulting  figure  would  be  so  small  and  out  of  line  with 
quotations  of  other  market  figures.  Reversely,  radium 
is  quoted  per  milligram  and  not  per  ton. 

That  is,  custom  has  already  unconsciously  assigned 
roughly  adjusted  weights  in  hitting  upon  the  units  respec- 
tively applied  not  only  to  silver,  coal,  hay,  radium,  but 
probably,  to  some  extent,  to  almost  everything.  I  am 
therefore  inclined  to  think  that  in  using  Formula  51  it  is 
belter  simply  to  add  the  newspaper  quotations  in  pounds, 
ounces,  tons,  yards,  etc.,  indiscriminately  rather  than  to 
reduce  them  to  one  unit.  This  reduction  is  based  on  the 
misconception  that  economic  weighting  is  a  physical 
matter. 

Nevertheless,  custom  has  not  done  its  job  well.  The 
same  substance  is  very  inconsistently  quoted  according 
to  its  various  stages  of  manufacture.  Cattle  per  head 
and  beef  per  pound  give  weights  for  Formula  51  widely 
different.  Iron  per  ton,  copper  per  pound,  pig  iron  per 
ton,  and  tin  plates  per  hundredweight  are  out  of  tune. 
Formula  51,  therefore,  unless  helped  out  by  judicious 
guessing  will  be  apt  to  play  freakish  tricks  upon  the 
user.  Sometimes,  in  fact,  unless  there  be  some  exercise 


WHAT  SIMPLE  INDEX  NUMBER  IS  BEST?    209 

of  judgment,  it  would  be  difficult  to  say  exactly  how  51 
is  to  be  interpreted ;  whether,  for  instance,  cotton  is  to 
be  entered  per  bale  or  per  pound,  its  quotation  being 
expressible  both  ways.  Formula  51  is  the  only  formula 
among  all  the  134  where  there  is  any  such  ambiguity. 
All  other  formulae  give  the  same  results  whether  cotton 
is  measured  in  pounds  or  bales. 

§  5.  Discarding  Formula  41  and  Possibly  Formula  31 

as  Freakish 

Quite  as  appropriate,  although  in  a  different  way,  is 
the  term  freakishness  as  applied  to  the  mode  and,  in  less 
degree,  to  the  median.  While  Formula  51  is  too  respon- 
sive to  changes  in  the  things  to  be  compared,  41  and  31 
are  less  responsive  than  the  other  formulae  to  the  influence 
of  change  in  any  individual  term. 

The  fatal  weakness  of  the  mode  (which  is  to  some 
extent  shared  by  the  median)  is  that  the  process  by  which 
it  is  calculated  gives  undue  influence  to  the  few  price 
relatives  which  happen  to  lie  together  in  its  vicinity,  and 
gives  practically  no  voice  at  all  to  the  rest  of  the  price 
relatives. 

Thus,  in  our  regiment  of  soldiers  where  we  found  the 
modal  height  to  be  about  5  feet  9|  inches,  this  figure 
would  still  be  the  mode  even  if  each  of  the  soldiers  taller 
than,  say,  5  feet  10  inches  were  replaced  by  a  new  soldier 
a  foot  taller  than  his  predecessor!  Reversely,  the 
shorter  men  have  practically  no  voice  in  determining  the 
mode.  The  modal  soldier  is  thus  not  a  fair  representative 
of  the  whole  regiment  because  most  of  the  soldiers  may 
be  taller  or  shorter  without  making  any  difference  to  the 
mode,  just  as  a  congressman  is  not  a  fair  representative 
of  his  district  when  chosen  by  a  clique. 

Where  the  number  of  price  relatives  is  small  the  mode 


210         THE  MAKING  OF  INDEX  NUMBERS 

is  particularly  haphazard.  With  a  large  number,  the  dis- 
tribution assumes  some  regularity,  and  the  mode  becomes 
more  significant.  Therefore  the  mode  cannot  properly 
be  used  unless  the  number  of  items  is  great,  and  then  it 
should  be  thought  of  as  only  a  rough  approximation. 
For  this  reason  it  is  practically  never  worth  while  to  use 
the  mode  as  an  index  number.  It  was  (with  some  re- 
luctance) included  in  my  list  because  it  has  been  dis- 
cussed in  connection  with  index  numbers,  and  because 
it  serves  as  a  foil  in  our  comparisons. 

§  6.   Freakishness  of  Simple  Median 

The  simple  median  is  much  more  nearly  representative 
of  all  the  price  relatives  than  is  the  mode,  and  yet  much  less 
representative  than  the  other  simple  index  numbers. 
Any  particular  soldier  in  the  regiment  could  be  taken  out 
and  replaced  elsewhere  by  another,  taller  or  shorter, 
without  displacing  the  median,  so  long  as  this  change  in 
height  of  the  particular  soldier  did  not  send  him  to  the 
other  side  of  the  median.  All  of  the  soldiers  standing  on 
one  side  of  the  middle  soldier  (say  the  shorter  side)  could 
be  replaced  by  still  shorter  soldiers,  even  dwarfs,  without 
changing  the  median  in  the  least.  Or  they  could  all  be 
replaced  by  taller  soldiers  up  to  the  middle  soldier's  height 
without  depriving  him  of  his  median  character  as  repre- 
sentative of  the  regiment.  Likewise,  on  the  tall  side, 
all  the  soldiers  could  be  replaced  by  giants  or  all  could 
shrink  to  the  median,  without  changing  the  latter.  In 
short,  the  median,  like  the  mode,  is  insensitive  or  unrespon- 
sive. Every  other  index  number,  such  as  the  arithmetic 
or  the  geometric,  would  faithfully  register  some  effect  of 
any  change  in  the  regiment,  however  slight.  The  extreme 
end  soldiers,  exactly  like  those  nearer  the  center,  have  some 
voice  and  influence  in  determining  the  average  height. 


WHAT  SIMPLE  INDEX  NUMBER  IS  BEST?  211 


If  one  of  them  grows  even  a  quarter  of  an  inch,  the  average 
will  be  affected.  The  mode  and  median,  on  the  other  hand, 
are  not  sensitive  barometers  but  creaky  weathervanes 
which  seldom  change,  and  when  they  do,  they  change  by 
jumps. 

If,  then,  we  are  justified  hi  excluding  Formula  51,  on 
account  of  its  freakish  weighting,  and  41  and  31  on  account 
of  their  freakish  insensitiveness,  we  have  left  out  of  our 
original  seven,  only  two  index  numbers,  viz.,  101  and 
21,  i.e.  the  arithmetic-harmonic  and  the  geometric. 
These  two  agree  very  closely,  so  that,  so  far  as  accuracy 
is  concerned,  there  is  nothing  to  choose  between  them. 
This  close  agreement  is  shown  in  the  following  table : 


FORMULA 
No. 

PRICES  —  FIXED  BASE 

1914 

1915 

1916 

1917 

1918 

21 

95.77 

96.79 

121.37 

166.65 

180.12 

101 

95.75 

96.80 

121.38 

166.60 

179.09 

§  7.   Doubt  as  to  Formula  31  vs.  Formula  21 

The  above  conclusion,  that  the  geometric  (or  its  equal, 
Formula  101),  is  the  best,  has  been  reached,  however, 
only  on  the  assumption  that  simple  weighting  is  proper 
weighting,  an  assumption  which  we  know  is  not  correct. 
In  the  absence  of  available  weights  we  are  sometimes 
forced  to  use  simple  or  equal  weighting  but  we  are  never 
justified  in  assuming  that  it  is  really  the  best  weighting. 
On  the  contrary,  we  must  assume  that  this  weighting 
contains  unknown  errors.  It  will  usually  be  found, 
when  the  true  weights  are  revealed,  that  the  simple 
weighting  was  not  only  erratic  but  so  erratic  as  to  de- 
serve to  be  called  freakish.  In  view  of  this  fact,  we  cannot 
yet  close  the  argument  and  give  judgment  to  the  geometric 
as  against  the  median.  If  the  commodities,  reckoned 


212         THE  MAKING  OF  INDEX  NUMBERS 

by  simple  weighting  as  though  they  were  of  equal  impor- 
tance, are  really  of  very  unequal  importance,  the  geometric 
may,  from  its  very  sensitiveness,  be  more  distorted  by  the 
false  weighting  than  the  median  by  its  insensitiveness. 

The  only  way  to  settle  the  question  whether,  in 
actual  fact,  the  simple  geometric  or  the  simple  median 
gives  the  closer  approximation  to  the  result  obtained  by 
proper  weighting,  is  actually  to  compare  these  three 
statistically.  This  will  be  done  in  the  next  chapter  with 
interesting  results. 

At  this  point  we  are  merely  justified  in  concluding  that 
if  the  simple  weighting  does  not  happen  to  be  too  erratic, 
the  geometric  (or  the  practically  coincident  Formula  101) 
is  the  best  formula  of  the  seven  considered  in  this  chapter. 


CHAPTER  XI 
WHAT  IS  THE  BEST  INDEX  NUMBER? 

§  1.  Introduction 

AT  the  beginning  of  the  last  chapter,  we  set  ourselves 
two  problems :  first,  to  find  the  best  simple  index  number, 
which  means  best  on  the  assumption  that  we  lack  the  full 
data  needed  for  weighting,  and,  second,  assuming  all 
needed  data  to  be  supplied,  to  find  the  very  best.  In  the 
last  chapter  we  took  up  the  first  problem.  We  are  now 
ready  to  study  the  second  (and,  incidentally  to  add  to 
our  conclusions  concerning  the  first). 

Let  us  assume,  then,  that  we  have  accurate  and  complete 
data  both  as  to  prices  and  quantities  and,  therefore, 
values.  The  specific  question  to  be  answered  in  this 
chapter  is :  What  formula  for  the  index  number  of,  say, 
prices  is  the  most  accurate  ? 

§  2.  Discarding  All  Simples  and  Their  Derivatives 

We  may  begin  by  excluding  not  only  all  simple  index 
numbers  but  all  of  their  derivatives.  Such  derivatives  are 
mongrels,  almost  contradictions  in  terms.  As  we  have 
seen,  a  simple  index  number  has  as  its  excuse  for  existence 
a  supposed  lack  of  available  weights.  Yet  we  have 
rectified  our  simple  index  numbers  by  Test  2,  although 
to  use  Test  2  presupposes  a  knowledge  of  weights.  Of 
course,  if  we  really  have  a  knowledge  of  these  weights  we 
should,  as  previously  pointed  out,  use  that  knowledge  at  the 
outset,  and  start  off  with  weighted  index  numbers.  No  one 

213 


214          THE  MAKING   OF  INDEX  NUMBERS 

could  argue  that  we  should  get  the  best  results  by  starting 
with  a  bad  index  number,  and  then  trying  to  reform  it 
by  the  processes  of  rectification. 

The  rectifications  of  simple  index  numbers,  therefore, 
are  mere  curiosities  to  show  how  far  the  faults  of  a  bad 
start  can  be  overcome  later.  The  results  will  be  considered 
at  the  proper  stage ;  but,  at  present,  in  searching  for  the 
most  accurate  index  number  possible,  we  must  rule  out  not 
only  all  simples,  but  all  then-  derivatives,  i.e.  then-  antithe- 
ses and  their  rectifications,  on  the  principle  that  we  should 
not  expect  to  "  make  a  silk  purse  out  of  a  sow's  ear." 

§  3.  Discarding  All  Modes  and  Medians  and 
Their  Derivatives 

We  have  just  ruled  out  group  "  S,"  the  simples.  We 
next  rule  out  group  "  M,"  the  modes  and  medians  (so 
far  as  they  have  not  already  been  ruled  out  by  being  in 
group  "  S  ")•  Previously,  in  discussing  the  mode  and 
median  types  of  index  number,  we  saw  that  they  were 
freakish  in  that  they  were  unresponsive  to  the  influence  of 
small  changes  in  the  terms  averaged.  On  this  account 
they  are  clearly  less  fitted  than  the  other  index  numbers 
to  provide  a  refined  barometer.  All  that  we  need  to  add 
here  is  that  this  freakishness  holds  true  of  the  weighted 
modes  and  medians  as  well  as  of  the  simple  modes  and 
medians.  In  fact,  not  only  are  the  mode  and  median  apt, 
so  to  speak,  to  fall  accidentally  into  the  clutches  of  a  few 
of  the  price  relatives  instead  of  being  equally  in  the  hands 
of  all,  but  the  weighted  mode  and  weighted  median  are  apt 
to  fall  accidentally  into  the  clutches  of  a  single  large  weight 
or  a  very  few  large  weights.  If  one  or  two  price  relatives 
near  the  middle  of  the  range  of  price  relatives  happen  to 
have  large  weights  they  are  apt  to  control  the  mode  or 
median  absolutely.  When  the  index  number  is  thus 


WHAT  IS  THE  BEST  INDEX  NUMBER?      215 

captured  no  ordinary  change  in  the  price  relative  can  dis- 
lodge it.  It  is,  so  to  speak,  "  stuck."  And  when  a  big 
enough  change  does  dislodge  it,  it  simply  jumps  into 
another  such  situation.  The  weighted  mode  is  thus  almost 
a  one-chance  proposition,  staking  everything,  perhaps,  on 
whether  or  not  some  one  commodity  with  a  monstrous 
weight  happens  fairly  to  represent  the  rest  in  its  price 
changes  —  the  chances  being,  naturally,  against  it,  In 
using  the  mode  we  almost  "  put  all  our  eggs  in  one  bas- 
ket." It  is  doubtful  whether  a  weighted  mode  (or  perhaps 
even  a  weighted  median)  is  a  better  barometer  than  a 
simple  mode  (or  simple  median),  especially  where  there 
are  only  a  few  commodities  involved. 

Because  of  this  characteristic  of  the  mode,  its  inertness, 
the  modes,  Formulae  143  and  145,  even  though  "  rectified  " 
by  Test  1  (i.e.  by  splitting  the  difference  between  43  and 
49,  and  between  45  and  47,  where  there  are  no  observable 
differences  to  split),  gam  no  real  improvement  in  accuracy. 

The  only  real  improvement  in  the  modes  effected  by  a 
"  rectification  "  comes  through  Test  2.  The  numerator 
of  the  factor  antithesis  is  the  value  ratio,  and  in  the  value 
ratio  every  element,  p  and  q,  has  a  voice.  But  this  kind 
of  rectification  has  power  to  correct  only  a  small  part  of 
the  freakishness  of  the  original.  And  it  may  be  balked 
in  accomplishing  even  a  partial  correction ;  for  the  denomi- 
nator of  the  factor  antithesis  (being  simply  another 
mode  —  of  quantities  instead  of  prices)  also  contains 
freakishness,  and  this  may  operate  in  either  direction. 
The  only  gain  is  that,  instead  of  (practically)  a  one-chance 
proposition,  we  now  have  a  two-chance  proposition. 

In  view  of  what  has  been  said  it  is  not  surprising  that  the 
modes  (and  to  some  extent  the  medians)  are  found  to  be  out 
of  tune  with  the  other  index  numbers,  sometimes  far  above 
and  sometimes  far  below,  without  rhyme  or  reason. 


216 


THE  MAKING  OF  INDEX  NUMBERS 


§  4.  Possible  Improvement  by  Increasing  the 
Number  of  Commodities 

This  freakishness  of  the  modes  (and  of  the  medians)  can, 
of  course,  be  lessened  by  including  a  large  number  of 
commodities  just  as  any  other  index  number  can  be 
improved  somewhat  in  the  same  way.  By  taking  a  very 
large  number  of  commodities,  we  could  perhaps  make  the 
rectified  weighted  modes  and  medians  approximately 
coincide  with  the  middle  tine  of  our  fork.  Unfortunately, 
we  have  no  data  for  testing  this  hypothesis  and  the  simple 
mode,  as  given  by  Wesley  C.  Mitchell,  for  the  1437  com- 
modities studied  by  the  War  Industries  Board,  is  as  far 
out  of  tune  with  the  other  types  of  index  numbers,  say 
the  simple  geometric,  as  is  the  simple  mode  of  our  36 
commodities. 

The  mode,  in  the  two  cases,  lies  above  (+)  or  below  (— ) 
the  simple  geometric  as  follows : 

TABLE  23.    EXCESS  OR   DEFICIENCY  OF  SIMPLE  MODE  OF 
PRICE  RELATIVES 

(In  per  cents  of  simple  geometric) 


No.  OF  COM- 
MODITIES 

1914 

1915 

1916 

1917 

1918 

36 

+2.08 

+  1.03 

-10.74 

-19.16 

+5.56 

1437 

+  1.52 

-5.93 

-19.75 

-14.64 

-10.53 

Thus,  irrespective  of  the  number  of  commodities, 
it  will  be  seen  that,  whereas  the  arithmetic  (as  pre- 
viously shown)  always  lies  above  the  geometric  and  the 
harmonic  always  below,  the  mode  is  above  and  below  about 
equally  often,  being  above  in  four  of  the  ten  cases  and  below 
in  six.  In  the  long  run  we  may  expect  this  approximate 
equality  to  be  more  perfect.  In  fact  it  is  absolutely 
perfect  if  we  always  take  into  account  backward  as  well  as 


WHAT  IS  THE  BEST  INDEX  NUMBER?      217 


forward  index  numbers,  for  if  the  forward  (or  backward) 
mode  is  above  the  geometric  the  backward  (or  forward) 
must  be  below  it.1  That  is,  the  mode  has  no  inherent 
tendency  to  lie  either  above  or  below  the  geometric. 
Either  is  equally  likely,  although  there  is  always  a  likeli- 
hood of  deviating  widely  —  freakishness. 

Exactly  the  same  discussion  applies  to  the  median 
except  that  the  freakishness  is  less.  For  the  simple 
medians  we  have : 

TABLE  24.     EXCESS  OR  DEFICIENCY  OF    SIMPLE  MEDIAN 
OF  PRICE  RELATIVES 

(In  per  cents  of  simple  geometric) 


No.  OF  COM- 
MODITIES 

1914 

1915 

1916 

1917 

1918 

36 

+3.84 

+1.84 

-  2.11 

-1.70 

+6.00 

1437 

-  .01 

-5.66 

-11.02 

-6.40 

-  .64 

Comparing  Table  24  with  Table  23,  it  will  be  observed 
that  the  median  and  mode  usually  jump  together,  first, 
on  one  side  of  the  geometric,  and  then  on  the  other ;  but 
the  median  usually  jumps  less  than  the  mode,  thus  lying 
between  the  mode  and  geometric.  The  average  ratio  of 
the  two  deviations  (those  of  the  mode  and  median 
from  the  geometric)  is  2.5  in  the  case  of  the  36  commod- 
ities and  2.2  in  the  case  of  the  1437  commodities. 

It  is  noteworthy  that  the  mode  and  median  seem  to  be 
below  the  geometric  when  prices  are  rapidly  rising. 
Whether  this  is  usually  the  case  and,  if  so,  why,  I  do  not 
know.  In  this  particular  case,  it  may  be  partly  accounted 
for  by  price  fixing  preventing  many  commodities  from 
rising  as  much  as  they  otherwise  would,  so  that  those  com- 
modities which  do  rise  inordinately  raise  the  geometric 
but  scarcely  affect  the  mode  or  median. 

1  See  Appendix  I  (Note  to  Chapter  XI,  §  4). 


218         THE  MAKING  OF  INDEX  NUMBERS 

While  the  freakishness  of  the  mode  and  median  can  prob- 
ably be  reduced  by  introducing  large  numbers,  it  cannot 
be  eliminated  altogether.  Under  all  circumstances  these 
index  numbers  are  lame  and  limping,  as  compared  with 
the  other  four  types.  I  have  estimated  very  roughly 
on  the  basis  of  the  data  above  mentioned  and  the  law  of 
distribution  of  chances  that,  for  a  large  number  of  com- 
modities, say  100,  the  rectified  mode,  Formula  343,  would 
keep  almost  always  within  two  per  cent  of  the  middle 
tine.  In  the  present  case  of  36  commodities,  it  is  ten  per 
cent  off  the  track  for  1917,  although  for  the  other  years 
it  is  usually  within  three  per  cent.  And  the  rectified 
median  is  within  two  per  cent  even  in  the  case  of  our  36 
commodities.  With  100  commodities  it  would  doubtless 
agree  still  more  closely. 

§  5.   Discarding  All  "  Biased  "  Index  Numbers 
Leaves  Only  the  Middle  Tine  (47  Formulae) 

Thus  far  in  our  search  for  the  most  accurate  index 
number,  we  have  eliminated  (1)  the  "  S  "  group,  i.e.  all 
simples  and  their  progeny,  and  (2)  the  "  M  "  group,  — 
all  medians  and  modes  and  their  progeny.  We  have 
found  these  index  numbers  "  freakish  "  or  "  haphazard/7 
the  first  group  because  constructed  from  badly  (that  is, 
evenly)  weighted  material,  and  the  second  because  so 
largely  insensitive  to  changes  in  the  individual  prices  and 
quantities. 

As  we  have  seen,  the  rest  of  the  index  numbers  do  not 
vary  at  random  but  naturally  group  themselves  into  the 
five  classes  shown  by  the  five-tined  fork.  That  is,  they 
differ  from  one  another  not  by  even  gradations  but  by 
definite  intervals.  The  causes  for  this  grouping  we  have 
already  investigated  and  expressed  by  the  term  "  bias  " 
to  represent  a  distinct  tendency  or  "  list  "  in  a  particular 


WHAT  IS  THE  BEST  INDEX  NUMBER?      219 

direction.  We  now  eliminate  all  biased  index  numbers 
(classes  2+,  !  +  ,!  —  ,  2  —  ),  viz.,  all  in  the  two  upper  and 
two  lower  tines,  leaving  only  the  "0"  or  unbiased  class 
for  further  consideration. 

§  6.   Selecting  from  the  47  Formulae,  the  13 
Satisfying  Both  Tests 

There  are  47  distinct  formulae  represented  in  this  middle 
tine.  Even  if  we  proceeded  no  further  we  would  have 
reached  an  important  conclusion  —  even  a  startling  con- 
clusion. These  47  formulae  agree  more  closely  than  the 
standards  of  ordinary  statistical  practice  require!  We 
may  say,  therefore,  that,  if  we  merely  exclude  formula  / 
obviously  freakish  or  biased,  all  the  rest  agree  with  each 
other  well  enough  for  ordinary  practical  purposes  ! 

But  we  may  go  still  further  in  our  search  for  accuracy. 
Among  these  47  approximately  agreeing  formulas  there  are 
two,  53  and  54,  which,  while  free  from  bias,  are  not  free 
from  joint  error.  For  instance,  Formula  53  forward  times 
Formula  53  backward  does  not  give  unity  but  sometimes 
a  little  more  and  sometimes  a  little  less,  revealing  a  slight 
joint  error  in  the  two  applications  of  53 ;  and  so  also  of  54. 
In  short,  Formulae  53  and  54  fail  to  obey  Test  1  as  also 
they  fail  to  obey  Test  2.  The  same  is  true  of  6053,  7053, 
8053,  and  8054.  Ruling  these  out,  we  have  left  41  for- 
mulae all  of  which  obey  at  least  one  test.  But  from  these 
we  may  eliminate,  as  not  obeying  both  tests,  Formulae  107, 
108,  109,  110,  1103,  1104,  123,  124,  125,  126,  1123,  1124, 
1153,  1154,  2153,  2154,  3153,  3154,  4153,  4154,  207,  209, 
213,  215,  223,  225,  227,  229.  We  now  have  left,  as 
obeying  both  tests,  13  formulae  from  which  to  choose  the 
best,  namely,  307,  309,  323,  325,(3^j  1303,  1323,  1353, 
2353,  3353,  4353,  5307,  5323. 

The  argument  here  is  not  that  every  one  of  these  so  far 


220         THE  MAKING  OF  INDEX  NUMBERS 

surviving  formulae  is  better  than  all  those  eliminated, 
for  we  shall  find  that  this  is  not  quite  true,  but  that  each 
of  the  excluded  formulae,  failing  in  one  or  the  other  of  the 
two  tests,  is  necessarily  surpassed  by  some  at  least  of  the  13. 
Thus  Formula  109,  failing  in  Test  2,  must  be  adjudged 
inferior  to  its  own  rectification,  309,  which  meets  both 
tests,  even  if  it  (Formula  109)  happens  to  be  superior 
to  some  other  of  the  inner  circle  of  13,  say,  1303 ;  and  we 
may  conclude  that  8053  and  8054  are  inferior  to  their  own 
rectification  (which  is  353)  without  concluding  that  they 
are  inferior  to  some  other  of  the  13,  say,  to  309. 

In  other  words,  out  of  the  47  best  formulae  we  are  select- 
ing, not  necessarily  the  best  13,  but  the  13  which  we  know 
must  include  the  best  one  of  all.  In  still  other  words, 
while  we  have  no  reason  to  think  that  each  of  these  13  is 
superior  to  all  the  34  excluded,  we  do  have  good  reason  to 
believe  that  each  of  these  34  is  inferior  to  some  one  of  the  13. 

§  7.   Selecting  Formula  353  as  the  "  Ideal  " 

We  have  still  to  choose  from  the  surviving  13,  although 
their  agreement  is  now  close  —  far  closer  than  practically 
required.  Here  the  argument  changes  and  becomes  much 
less  definite  and  sure.  We  can  no  longer  appeal  to  the  two 
tests  as  a  means  of  further  sifting ;  for  all  the  13  formulae 
obey  both  these  tests  perfectly.  But  we  can  still  find 
reasons  for  preferring  one  formula  to  another.  We  can 
prefer  the  crossed  formula  to  the  cross  weight  formula 
and  their  derivatives  (except  Formula  2353,  reserved 
for  later  consideration),  thus  excluding  1303,  1323,  1353, 
3353,  4353,  and  leaving  only  the  eight  formulae :  307, 309, 
323,  325,  353,  2353,  5307,  5323. 

This  exclusion  is  based  on  the  consideration *  that  the 
cross  weight  formulae  fail  to  insure  a  middle  course  between 
1  As  shown  in  Appendix  I  (Note  to  Chapter  VIII,  §  10). 


WHAT  IS  THE  BEST  INDEX  NUMBER?      221 

the  original  formulae  whose  weights  are  crossed.  They 
seem  slightly  erratic  as  compared  with  the  rest.  Again, 
on  the  principle  that  two  equally  promising  estimates  or 
measures  may  probably  be  unproved  in  accuracy  by  taking 
their  average,  307  and  309  may  next  be  excluded  in  favor 
of  their  cross,  5307 ;  and,  likewise,  323  and  325,  in  favor 
of  their  cross,  5323. 

This  leaves  the  four  formulae,  353,  2353,  5307,  5323. 
From  these  four,  all  practically  coinciding,  I  should  be 
inclined,  if  forced  to  choose,  first,  to  drop  2353  in  favor  of 
353  on  the  theory  that  weight  crossing  of  any  kind  is 
probably  not  as  accurate  a  splitter  of  differences  as  formula 
crossing.  This  leaves  the  three  formulae,  353,  derived 
from  aggregatives ;  5307,  derived  from  arithmetics  and 
harmonics;  and  5323,  derived  from  geometries.  From 
these  I  am  inclined  next  to  eliminate  5307  on  the  ground 
that  it  descends  from  ancestors  (7,  8,  9,  10,  13,  14,  15,  16) 
far  wider  apart  than  does  353  or  5323.  There  seems  more 
chance  of  error  in  using  figures  wide  apart  than  in  using 
those  close  together.  If  we  must  prefer  one  of  the  two 
remaining  formulae  (353  and  5323)  to  the  other  I  would 
drop  5323  for  the  same  reason. 

Thus  Formula  353,  derived  from  aggregatives,  remains  \\ 
to  take  the  first  prize  for  accuracy.     But  I  should  not 
quarrel  with  those  who  would  divide  the  prize  with  2353, 
5307,  or  5323,  especially  the  last. 

§  8.   Other  Arguments  for  Formula  353 

Our  whole  argument  has  hitherto  been  on  the  score  of 
accuracy.  If  we  add  the  consideration  of  algebraic 
simplicity,  the  superiority  of  Formula  353  over  all  its 
rivals  is  evident,  and  very  marked.  As  to  ease  and 
rapidity  of  computation  (of  which  I  shall  speak  more 
fully  later)  353  is  immensely  superior  to  all  its  12  rivals, 


222         THE  MAKING  OF  INDEX  NUMBERS 

though  some  excellent  formulae  outside  of  the  13  are  still 
more  rapid,  as  we  shall  see. 

Hitherto  no  use  has  been  made  of  the  argument  that 
formulae  of  widely  different  nature  are  likely  to  be  accu- 
rate if  they  agree  with  each  other.  Every  formula  was 
given  consideration  independently  and  on  its  merits. 
Thus,  Formula  9  was  condemned,  not  on  the  ground  that 
it  gave  results  higher  than  353  and  the  other  middle-of- 
the-roaders,  but  on  the  ground  that,  if  twice  applied,  once 
forward  and  again  backward,  it  gave  a  result  greater 
than  unity  so  that  at  least  one  of  its  two  applications 
was  too  large.  We  could  thus  prove  bias  without  any 
comparison  with  other  formulae.  Likewise  41  and  43 
were  condemned  as  freakish,  not  because  they  differ 
so  greatly  from  other  formulae,  but  because  they  fail  to 
respond  to  most  of  the  changes  which  they  aim  to  average. 

And  yet  as  we  have  proceeded,  step  by  step,  we  could 
not  fail  to  notice  that  the  good  formulae  give  very  similar, 
and  the  bad  formulae,  very  dissimilar,  results,  and  that 
the  good  agree  in  results  despite  wide  differences  of 
method.  And  now  that  we  have  completed  the  original 
line  of  argument,  we  may  confirm  it  strikingly  by  citing, 
as  new  and  internal  evidence,  these  similarities  and  dis- 
similarities. The  formulae  which  we  condemned  as  up- 
ward biased  (on  the  ground  of  comparison  only  with  them- 
selves, reversed  hi  direction),  we  now  find  do  actually  give 
higher  results  than  353  and  its  peers  or  next  bests, 
the  divergence  for  the  doubly  biased  formulae  being  about 
double  the  divergence  of  the  singly  biased  formulae; 
and  similarly  as  to  the  downward  biased. 

The  only  qualifications  to  this  statement  are  such  as 
merely  further  confirm  what  has  been  found.  Thus  the 
simples,  modes,  medians,  and  their  derivatives,  which, 
on  general  grounds,  were  condemned  as  very  erratic, 


WHAT  IS  THE  BEST  INDEX  NUMBER?      223 

behave  peculiarly  relative  to  353  and  the  other  foremost 
formulae,  and  thereby  again  justify  the  term  "  freakish. " 
Thus  all  the  formulae  shown  to  be  bad  independently 
are  found  also  to  be  bad  comparatively,  —  that  is,  as 
judged  by  their  departures  from  the  very  good  formulae. 

Finally,  the  formulae  which  we  found,  by  studying  each 
one  by  itself,  to  be  good,  because  free  from  bias  and  freak- 
ishness,  are  also  found  to  be  good  as  judged  by  each  other. 
That  is,  they  all  agree  amazingly  well,  constituting  the 
middle  tine  of  the  fork.  In  fact,  I  think  that  anyone 
who  had  not  followed  the  former  argument  but  who  should 
merely  examine  the  internal  evidence  of  agreement  and 
disagreement  would  reach  almost  exactly  the  same  con- 
clusions as  to  which  formulae  are  good  and  which  are  bad. 
At  any  rate  the  agreements  and  disagreements  between  the 
134  formulae  are,  without  a  single  exception,  consistent 
with  all  the  conclusions  reached  on  other  grounds. 

§  9.    Formulae  353  and  5323  Compared 

We  have  seen  that  Formulae  353  and  5323  present 
almost  equal  claims  to  be  true  barometers  of  changes  in 
prices  and  quantities.  But  their  results  do  not  tally 
absolutely,  as  Table  25  shows.1 

In  only  one  instance  do  the  two  methods  yield  precisely 
the  same  result  and  that  identity  would  doubtless  dis- 
appear if  we  were  to  carry  the  computation  one  decimal 
further. 

What  are  we  to  infer  from  these  disagreements  ?  Error 
there  must  be  but  we  have  no  warrant  for  saying  one  is 
"  absolutely  right  "  and,  therefore,  all  the  error  is  in  the 

1  For  the  purposes  of  this  comparison  Formula  353  might  have  been 
called  5353  (although  no  such  number  is  used  in  the  list),  for  just  as  5323 
is  descended  from  eight  index  numbers  (23,  24,  25,  26,  27,  28,  29,  30),  so 
may  5353  (i.e.  353)  be  regarded  as  derived  from  eight  (3,  4,  5,  6, 17, 18,  19, 
20). 


224 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  25.    TWO  BEST  INDEX  NUMBERS 
(1913  =  100) 


PRICES 

QUANTITIES 

Base 

Formu- 
la No. 

1914 

1915 

1916 

1917 

1918 

1914 

1915 

1916 

1917 

1918 

Fixed 
Chain 

353 
5323 

100.12 
100.13 

99.89 
99.87 

114.21 
114.09 

161.56 
161.59 

177.65 
177.67 

99.33 
99.32 

109.10 
109.11 

118.85 
118.92 

118.98 
118.96 

125.37 
125.35 

353 
5323 

100.  J2 
100.13 

100.23 
100.23 

\14.32 

114.45 
J 

162.23 
162.42 

178.49 
178.64 

99.33 
99.32 

108.72 
108.73 

118.74 
118.61 

118.49 
118.36 

124.77 
124.68 

other.  We  must  infer  just  what  Pierson  inferred,  that 
index  numbers  are  not  and  never  can  be  absolutely  pre- 
cise. There  is  always  a  fringe  of  uncertainty  surrounding 
them.  But,  while  index  numbers  can  never  quite  pretend 
to  rank  with  weights  and  spatial  measures  in  perfection  of 
precision,  Table  25  reveals  a  very  high  degree  of  precision, 
not  only  far  higher  than  skeptics  like  Pierson  imagined 
possible,  but  higher  even  than  believers  in  index  numbers 
had  supposed. 

Table  25  shows  that,  for  prices,  the  two  fixed  base  fig- 
ures for  1914  agree  within  about  one  part  in  10,000 ;  1915 
within  about  two  parts  in  10,000 ;  1916  within  about  one 
part  in  1000 ;  1917  within  about  one  part  in  5000 ;  and 
1918  within  about  one  part  in  9000 ;  while,  for  quantities, 
the  corresponding  degrees  of  agreement  are  substantially 
the  same:  one  part  in  10,000,  one  in  10,000,  one  in  1000, 
one  in  6000,  one  in  6000.  Turning  to  the  chain  index 
numbers  we  find,  for  prices  for  1915,  perfect  agreement 
as  far  as  calculated,  and  for  the  succeeding  years  one  in 
1000,  one  in  1000,  one  in  800 ;  while  for  quantities  the 
figures  are:  one  in  10,000,  one  in  10,000,  one  in  1000, 
one  in  1000,  one  in  1000. 

When  we  speak  of  two  magnitudes  as  agreeing  within 


WHAT  IS  THE  BEST  INDEX  NUMBER?      225 

one  part  in  1000  we  are  speaking  of  an  extremely  high 
degree  of  agreement.  The  agreement  is  as  close  as  that 
between  two  estimates  of  the  height  of  Washington 
Monument  which  differ  by  a  hand's  breadth,  or  two 
estimates  of  the  height  of  a  man  which  differ  by  a  four- 
teenth of  an  inch,  or  two  estimates  of  his  weight  which 
differ  by  two  ounces.  These  are  higher  degrees  of  preci- 
sion than  those  met  with  hi  the  measures  of  commodities 
sold  at  retail  and  than  most  of  those  met  with  in  whole- 
sale transactions.  They  are  comparable  even  with  many 
laboratory  measurements.  Thus,  I  learn  from  the  United 
States  Bureau  of  Standards  that  measures  of  volume  by 
glass  or  brass  containers  are  correct  only  to  one  part  hi 
5000  to  10,000.  The  best  portable  ammeter  measures 
electric  current  only  to  one  part  in  250,  and  the  best  port- 
able voltmeter  measures  voltage  to  only  one  part  in  500. 

When  we  consider  that  these  two  methods  of  reckoning 
an  index  number  by  Formulae  353  and  5323  are  wholly 
distinct,  that,  in  one,  the  processes  are  adding  and  divid-  \y 
ing,  and,  in  the  other,  they  are  multiplying  and  extract- 
ing roots,  it  seems  truly  marvelous  that  by  such  widely 
different  routes  we  should  be  led  to  almost  absolutely  the 
same  goal.  It  would  be  absurd  to  ascribe  the  agreement 
wholly  to  "  accident."  The  coincidences  are  too  numer- 
ous, even  without  recourse  to  the  agreements  with  other 
index  numbers  on  the  middle  tine.  We  cannot  escape 
the  conclusion  from  this  comparison  that  these  two  index 
numbers  -check  each  other  up  and  prove  each  other's  ac- 
curacy within  an  error  of  usually  less  than  one  part  in  1000. 

,  §  10.  The  "  Probable  Error  "  of  Formula  353 

We  may  now  cite  the  close  agreement  of  all  the  13  for- 
mulae which  satisfy  both  tests  and  are  also  free  of  the 
accusation  of  freakishness  (i.e.  are  not  descended  from 


226 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  26.    SELECTED  INDEX  NUMBERS 

(1913  =  100) 

FIXED  BASE 


FOR- 
MULA 
No. 

PRICES 

QUANTITIES 

1914 

1915 

1916 

1917 

1918 

1914 

1915 

1916 

1917 

1918 

307 

100.13 

99.78 

114.17 

161.04 

177.25 

99.31 

109.20 

118.89 

119.36 

125.65 

309 

100.17 

99.85 

114.25 

162.31 

178.44 

99.29 

109.13 

118.74 

118.43 

124.81 

323 

100.13 

99.89 

113.99 

161.90 

177.98 

99.31 

109.09 

119.09 

118.74 

125.14 

325 

100.12 

99.85 

114.19 

161.28 

177.35 

99.33 

109.13 

118.76 

119.19 

125.57 

353 

100.12 

99.89 

114.21 

161.56 

177.65 

99.33 

109.10 

118.85 

118.98 

125.37 

1303 

100.14 

99.88 

114.22 

161.75 

177.82 

99.32 

109.11 

118.84 

118.84 

125.25 

1323 

100.13 

99.90 

114.23 

161.70 

177.80 

99.32 

109.08 

118.84 

118.88 

125.26 

1353 

100.13 

99.89 

114.22 

161.71 

177.79 

99.33 

109.08 

118.85 

118.87 

125.26 

2353 

100.13 

99.89 

114.22 

161.60 

177.67 

99.32 

109.09 

118.84 

118.94 

125.35 

3353 

100.14 

99.90 

114.35 

161.94 

177.36 

99.31 

109.08 

118.81 

118.70 

125.57 

4353 

100.13 

99.92 

114.26 

161.78 

177.52 

99.32 

109.06 

118.80 

118.82 

125.46 

5307 

100.15 

99.82 

114.21 

161.67 

177.84 

99.30 

109.17 

118.81 

118.90 

125.23 

5323 

100.13 

99.87 

114.09 

161.59 

177.67 

99.32 

109.11 

118.92 

118.96 

125.35 

CHAIN  OF  BASES 


FOPMU- 

LANO. 

1914 

1915 

1916 

1917 

1918 

1914 

1915 

1916 

1917 

1918 

307 

100.13 

100.22 

114.56 

162.50 

178.42 

99.31 

108.74 

118.49 

118.30 

124.83 

309 

100.17 

100.22 

114.61 

162.76 

179.30 

99.29 

108.74 

118.44 

118.10 

124.21 

323 

100.13 

100.23 

114.45 

162.47 

178.69 

99.31 

108.73 

118.61 

118.32 

124.64 

325 

100.12 

100.23 

114.45 

162.36 

178.58 

99.33 

108.73 

118.62 

118.39 

124.71 

353 

100.12 

100.23 

114.32 

162.23 

178.49 

99.33 

108.72 

118.74 

118.49 

124.77 

1303 

100.14 

100.23 

114.40 

162.37 

178.99 

99.32 

108.72 

118.66 

118.39 

124.42 

1323 

100.13 

100.24 

114.65 

162.71 

179.05 

99.32 

108.72 

118.41 

118.14 

124.39 

1353 

100.13 

100.23 

114.33 

162.27 

178.45 

99.33 

108.72 

118.73 

118.46 

124.76 

2353 

100.13 

100.23 

114.32 

162.31 

178.58 

99.32 

108.72 

118.74 

118.43 

124.71 

3353 

100.14 

100.24 

114.28 

162.14 

178.39 

99.31 

108.71 

118.71 

118.48 

124.77 

4353 

100.13 

100.24 

114.38 

162.20 

178.46 

99.32 

108.71 

118.68 

118.51 

124.79 

5307 

100.15 

100.22 

114.59 

162.63 

178.86 

99.30 

108.74 

118.47 

118.20 

124.52 

5323 

100.13 

100.23 

114.45 

162.42 

178.64 

99.32 

108.73 

118.61 

118.36 

124.68 

WHAT   IS  THE  BEST   INDEX  NUMBER?       227 


simples,  modes,  or  medians).  Table  26  gives  these  13 
index  numbers  from  which  we  selected  Formula  353  as 
presumably  the  best. 

By  means  of  Table  26  we  can  further  address  our- 
selves to  the  problem  of  measuring  the  degree  of  accuracy 
of  Formula  353.  Critics  like  Pierson  have  cited  the 
disagreements  of  index  numbers,  which  they  mistakenly 
assumed  to  present  equal  claims  to  be  true  barometers  of 
price  changes,  as  evidence  that  index  numbers  in  general 
were  inaccurate.  Though  their  premises  were  wrong  their 
logic  was  right.  And  we  may  now  apply  it,  freed  from 
their  mistaken  premises.  We  may  apply  to  these  13 
barometer  readings  the  processes  of  the  theory  of  proba- 
bilities, and  compute  the  probable  errors.  We  shall 
assume,  at  the  outset,  that  all  the  13  have  equal  claims ; 
that  is,  we  shall  give  them  equal  weights  in  our  probability 
calculations.  This  is  conservative;  that  is,  it  will  tend 
to  exaggerate  the  probable  errors  of  the  best.  Table  27 
gives  the  probable  errors  as  so  calculated. 

TABLE  27.  PROBABLE  ERRORS1  OF  AN  INDEX  NUMBER 
OF  PRICES  OR  QUANTITIES  WORKED  OUT  BY  ANY  ONE 
OF  THE  13  FORMULAE  CONSIDERED  AS  EQUALLY  GOOD 
INDEPENDENT  OBSERVATIONS 

(In  per  cents  of  their  average) 


BASE 

1914 

1915 

1916 

1917 

1918 

Fixed 

.009 

.025 

.050 

.128 

.118 

Chain 

.009 

.006 

.069 

.079 

.104 

1  See  Appendix  I  (Note  to  Chapter  XI,  §  10). 


Thus  we  see  that  the  probable  error  of  any  of  the 
13  formulae  for  1914  was  .009  per  cent  to  be  added  to  or 
subtracted  from  the  index  number,  say,  100.12,  or  one 
part  in  10,000.  To  state  this  exactly,  assuming  all  of  the 


228         THE  MAKING  OF  INDEX  NUMBERS 

13  to  be  equally  likely  to  be  right,  the  error  of  any  one  of 
them  is  as  likely  as  not  less  than  one  part  in  10,000.  Simi- 
larly, the  probable  (or  as-likely-as-not)  error  in  1916  of 
the  fixed  base  figures  is  .05  per  cent  of  the  index  number 
—  about  one  part  in  2000. 

The  largest  error  in  a  single  index  number  is  that  for 
1917  relatively  to  1913.  That  is  the  index  number  has 
an  error  of  .128  per  cent,  or  about  one  part  in  800,  or  about 
one  eighth  of  one  per  cent.  We  may,  therefore,  be 
assured  that  Formula  353,  being  certainly  more  accurate, 
if  that  be  possible,  than  at  least  most  of  the  other  12, 
is  able  correctly  to  measure  the  general  trend  of  the  36  dis- 
persing price  relatives  or  quantity  relatives  within  less  than 
one  eighth  of  one  per  cent!  That  is,  the  error  in,  say,  For- 
mula  353,  probably  seldom  reaches  one  part  in  800,  or  a 
hand's  breadth  on  the  top  of  Washington  Monument,  or 
less  than  three  ounces  on  a  man's  weight,  or  a  cent  added 
to  an  §8  expense. 

The  above  estimate  of  one  eighth  of  one  per  cent  is  a 
maximum,  for  three  reasons :  (1)  it  is  the  maximum  of 
the  ten  figures  hi  the  above  table ;  (2)  the  above  table  is 
based  on  an  extraordinary  war-time  dispersion  which 
tends  to  magnify  the  disagreements  between  index  num- 
bers ;  and  (3)  many  of  the  13  formulae  treated  as  equally 
reliable  are  demonstrably  less  reliable  than  353.  To  re- 
place the  above  maximum  estimate  by  a  more  truly  repre- 
sentative one  is  not  easy  and  introduces  doubtful  consider- 
ations. Without  detailing  these,  I  shall  merely  say  that 
after  various  other  calculations  I  am  convinced  that  the 
probable  error  of  Formula  353  seldom  reaches  one  per  cent 
of  one  per  cent. 

Assuming  that,  for  practical  purposes,  a  precision  within 
one  per  cent  of  the  truth  is  ample,  we  see  that  any  first 
class  index  number  is  at  least  eight  times  as  precise  as  it 


WHAT  IS  THE  BEST  INDEX  NUMBER?      229 

needs  to  be.  Humanly  speaking  then,  an  index  number  is  t 
an  absolutely  accurate  instrument.  This  does  not,  of 
course,  have  any  reference  to  inaccuracies  hi  the  original 
data,  nor  to  inaccuracies  due  to  the  choice  of  data  in- 
cluded as  samples,  or  representatives  of  those  excluded. 
It  merely  means  that,  given  these  data,  the  index 
number  is  able  to  give  an  unerring  figure  to  express  their 
average  movement.  As  physicists  or  astronomers  would 
say,  the  "instrumental  error  "  is  negligible.  The  old  idea 
that  among  the  difficulties  in  measuring  price  move- 
ments is  the  difficulty  of  finding  a  trustworthy  mathe- 
matical method  may  now  be  dismissed  once  and  for 


§  11.  The  Purpose  to  Which   an  Index  Number  Is 
Put  Does  Not  Affect  the  Choice  of  Formula 

It  will  be  noted  that  Formula  353,  or  its  rivals,  has  been 
selected  as  the  best  on  very  general  grounds  of  a  formal 
character.  Consequently,  the  conclusions  are  as  general 
as  the  premises  from  which  we  started.  Whether  prices 
are  wholesale  or  retail,  for  instance,  obviously  does  not 
affect  the  choice  of  Formula  353  rather  than  1,  or  31,  or  9. 
For,  hi  either  case,  there  are  precisely  the  same  reasons 
for  selecting  a  formula  which  is  reversible  in  tune  or  factors 
and  for  selecting  a  formula  which  will  not  be  freakish, 
or  spasmodic,  in  its  findings. 

But  so  deeply  rooted  is  the  idea  that  various  purposes 
require  various  formulae  that  the  general  significance  of 
these  results  is  not  yet  acknowledged  by  many  of  the  stu- 
dents of  index  numbers.  I  must  reserve  for  a  separate 
article  specific  answer  to  those  who  have  rejected  the  con- 
clusion, when  first  briefly  stated  at  the  Atlantic  City  meet- 
ing of  the  American  Statistical  Association,  December, 
1920,  that  a  good  formula  for  one  purpose  is  a  good  formula 


230         THE  MAKING  OF  INDEX  NUMBERS 

for  all  known  purposes.     But  I  may  note  here  reasons 
alleged  for  rejecting  this  idea.     There  seem  to  be  three : 

(1)  There  is  the  idea  that  a  conflict  exists  between 
measuring  the  average  change  of  prices  and  measuring 
changes  in  the  average  level  of  prices.1 

(2)  There  is  the  idea  that  changes  in  the  aggregate  cost 
of  a  specified  bill  of  goods  or  regimen,  as  implied  in  aggre- 
gative  index   numbers,    is   appropriate   only   for   retail 
trade  —  despite  the  fact  that  Knibbs,  the  chief  protago- 
nist "of  this  concept,  applies  this  idea  of  aggregate  cost 
to  a  specified  list  of  wholesale  prices.     Of  course,  it  may 
be  applied  to  any  list  in  any  market.     What  is  the  custom 
in  the  case  has  nothing  to  do  with  the  accuracy  of  the 
procedure  as  a  mathematical  method. 

(3)  There  is  the  idea  that  the  character  of  the  dis- 
tribution of  price  relatives  about  the  mode  or  other  mean 
prescribes  the  choice  of,  say,  the  arithmetic  or  geometric 
type.    This  argument  defeats  itself  through  the  reversal 
process ;  any  asymmetry  displayed  (on  the  ratio  chart, 
at  least)  in  the  distribution  of  the  relatives  taken  forward 
is  reversed  when  we  have  to  consider  the  relatives  taken 
backward.2     If  the  arithmetic  be  adjudged  proper  for 
the  one  it  would  have  to  be  adjudged  improper  for  the 
other,  thus  leading  to  such  an  absurd  conclusion  as  that, 
in  calculating  the  price  level  of  London  relatively  to  New 
York,  the  arithmetic  index  number  is  appropriate,  but 
in  calculating  the  price  level  of  New  York  relative  to 
London  it  would  be  highly  improper!    Moreover,  if  we 
count  the  cases  in  both  directions  there  are,  of  course, 
as  many  cases  of  asymmetry  in  one  direction  as  in  the 
other.     It  follows  that,  in  the  long  run,  there  is  no  tend- 

1  This  is  discussed  in  Appendix  III  (on  ratio  of  averages  vs.  average  of 
ratios). 

2  As  pictured  in  Appendix  I  (Note  to  Chapter  XI,  §  11). 


WHAT  IS  THE  BEST  INDEX  NUMBER?      231 

ency  to  asymmetry  in  any  one  direction.1  Also  when  a 
large  number  of  relatives  are  used,  there  is  usually  little 
asymmetry  in  any  case.  The  opinion  to  the  contrary 
is  based  on  the  wrong  method,  usually  employed,  of 
plotting  on  an  arithmetic  scale  instead  of  the  ratio  chart 
used  in  this  book. 

But,  from  a  practical  standpoint,  it  is  quite  unnecessary 
to  discuss  the  fanciful  arguments  for  using  "  one  formula 
for  one  purpose  and  another  for  another,"  in  view  of  the 
great  practical  fact  that  all  methods  (if  free  of  freakish- 
ness  and  bias)  agree!  Unless  someone  has  the  hardihood 
to  espouse  bias  or  freakishness  for  some  "  purpose," 
whatever  formula  he  advocates  will  insist  on  coinciding 
with  whatever  formula  anyone  else  advocates.  The 
notion  that  the  aggregative  is  appropriate  for  the  cost  of 
living,  and  the  geometric  for  a  wholesale  price  level,  and 
the  arithmetic  for  something  else,  becomes  futile.  For 
if  we  admit  that  in  each  case  the  rectified  forms  are  to  be 
used,  we  shall  find  that  the  rectified  aggregative  (Formulse 
353,  1353,  2353,  3353,  4353),  the  rectified  geometric 
(323,  325,  1323,  5323),  and  the  rectified  arithmetic  (307, 
309, 1303,  5307),  all  agree  ten  times  as  closely  as  is  required 
for  any  purpose  whatever ! 

The  basic  reason  for  misunderstanding  on  this  subject  is 
failure  to  take  into  account  bias  and  reversibility  in  time. 
So  long  as  the  very  bad  Formulae  1,  9001,  21,  9021,  31, 
51,  9051  are  used,  no  wonder  writers  on  index  numbers 

1  Asymmetrical  distribution  is  often  characteristic  in  other  fields  than 
index  numbers,  e.g.  human  heights  or  weights.  (See  Macalister,  "Law  of 
the  Geometric  Mean,"  Proceedings  of  the  Royal  Society,  1879.)  But  in 
such  cases  there  is  no  reversibility.  The  items  averaged  are  not  ratios. 
In  the  case  of  a  skull  index,  on  the  other  hand,  the  ratio  of  length  to 
breadth  may  be  reversed  as  breadth  to  length  and  is  analogous  to  index 
numbers  which  are  ratios  of  prices  to  prices  or  quantities  to  quantities. 
Ratios  are  essentially  double  ended  and  produce  their  own  symmetry,  by 
reversal  in  one  form  or  another. 


232         THE  MAKING  OF  INDEX  NUMBERS 

seek  fanciful  reasons  for  using  one  of  these  mutually 
conflicting  formulae  for  one  purpose  and  another  for 
another.  But  as  soon  as  it  is  seen  that  the  weighted  index 
numbers  of  all  these  types  need  rectification,  — that  there 
is  no  more  justification  for  using,  for  instance,  an  arithmetic 
forward  than  backward,  and  that,  therefore,  it  should  be 

Weighted  Aggregatives  For  90  Raw  Materials 

War  Industries  Board  Statistics 
(Prices) 


12' 


!&  14  75  '/£  77  75 

CHAKT  45  P.  Showing  the  same  close  agreement  and  absence  of  bias 
of  Formulas  53  and  54  for  the  90  commodities,  as  was  found  in  the  case  of 
the  36  commodities  (see  Chart  39P,  top  tier) . 

rectified  before  being  used  at  all,  —  all  these  fanciful 
distinctions  and  arguments  fall  to  the  ground. 

A  year  ago  I  issued  a  friendly  challenge  to  those  who 
object  to  this  conclusion  to  supply  a  single  case  where 
Formula  353  should  not  be  used.  Several  have  tried  to 
supply  such  cases  but  without  success. 

It  is  clear  that  a  considerable  part  of  the  disagreement 
is  more  apparent  than  real  and  due  to  misunderstandings. 
Mitchell  gives  seven  purposes  requiring,  he  alleges,  dif- 


WHAT  IS  THE  BEST  INDEX  NUMBER?      233 

ferent  formulae.1  One  of  these  "  purposes  "  is  the  com- 
parison with  an  existing  series  of  index  numbers,  in  which 
case  the  formula  used  should  be  identical  with  that 
used  in  the  existing  series.  Naturally !  In  a  somewhat 
similar  way  I,  myself,  in  this  book,  have  found  use  for 
134  different  formulae  for  the  "  purpose  "  of  comparison. 
Another  of  Professor  Mitchell's  purposes  is  to  make  an 
index  number  which  the  common  man  can  understand. 
Of  course,  we  can  go  on  indefinitely  enumerating  such 
varieties  of  purpose.  Our  purpose  may  be  to  secure  the 

Weighted  Aggregatives  for  90  Raw  Materials 

War  Industries  Board  Statistics 
(Quantities) 

.54 

'53 

J5, ^^  \5% 

'15  '/4  75  7<£  77  75 

CHART  45Q.    Analogous  to  Chart  45P. 

cheapest  index  number.  Then  Formula  51  is  the  formula 
we  want.  Or  our  purpose  may  be  to  secure  the  most 
inaccurate.  One  of  the  modes  might  then  be  indicated. 
Formula  353  would  not  be  the  best  for  that  purpose ! 

I  had  assumed,  of  course,  that  there  was  at  least  this 
uniformity  of  "  purpose  "  :  that  by  the  best  index  number 
would  be  understood  the  index  number  which  was  the 
most  accurate  measure.  If  this  be  taken  for  granted,  353 
(or  any  of  the  30  or  more  others  which  give  the  same 
results)  seems  the  best  for  all  purposes  within  the  domain 
covered  by  index  numbers.  Whether  the  purpose  be  an 
index  number  of  prices,  or  quantities,  or  wages,  or  rail- 

1  Bulletin  No.  284,  United  States  Bureau  of  Labor  Statistics,  pp.  76,  78. 


234         THE  MAKING  OF  INDEX  NUMBERS 

road  traffic,  or  whether  the  index  number  is  to  measure  the 
value  of  money,  the  barometer  of  trade,  the  cost  of  manu- 
facturing, the  volume  of  manufacture,  the  same  varieties 
of  mathematical  processes  can  be  used  and  will  converge 
to  close  agreement,  —  that  is,  so  long  as  the  problem 
is  of  the  same  mathematical  form,  —  as  it  is  in  all  cases  I 
have  yet  met  with.1 

In  short,  an  index  number  formula  is  merely  a  statis- 
tical mechanism  like  a  coefficient  of  correlation.     It  is  as 

Formulae  55  and  54  Applied  To  Stock  Market 

(Prices) 

\5% 


16  17  18  19  20  2/ 

t92l 

CHART  46P.  Showing  the  same  closeness  of  agreement  and  absence  of 
bias  of  53  and  54  for  stock  market  prices. 

absurd  to  vary  the  mechanism  with  the  subject  matter  to 
which  it  is  applied  as  it  would  be  to  vary  the  method  of 
calculating  the  coefficient  of  correlation. 

§  12.   Comments  on  Formula  353  and  the 
Aggregative  s  Generally 

Formula  353  must  already  have  impressed  the  reader 
as  having  noteworthy  peculiarities  and  simplicities.  It 
is  formed  more  simply 2  than  any  other  formula  ful- 

1  See  Appendix  I  (Note  to  Chapter  IV,  §  10). 

2  To  see  how  much  simpler  353  really  is  than  any  other  formula  among 
the  12  rivals  for  accuracy  we  need  only  compare  it  with  the  next  in  sim- 
plicity, 2353,  as  follows : 


WHAT   IS  THE  BEST   INDEX  NUMBER?      235 

filling  the  two  tests,  being  obtained  merely  from  the  four 
magnitudes  2p0<Zo,  ^Pi^i>  2p0(?i,  £pi<7o-     The  same  four  are 

Formulae  55 and  54  Applied  To  Stock  Market 

(Quantities) 


I** 


16 


17 


HAY- 
1321 


/9 


20 


CHART  46Q.     Analogous  to  Chart  46P. 


used,  simply  in  different  order,  for  the  price  index  number 
and  the  quantity  index  number.1 

The  formula  fulfills  both  tests,  although  it  is  obtained 
by  only  one  crossing  of  antecedent  formulae.    That  one 


2353 


-V 


x 


x 


X  2Wo  +  2i)  Po  X 

The  reader  may  care,  for  curiosity,  to  write  out  some  of  the  still  more 
complicated  formulae  such  as  5323,  the  most  accurate  among  the  geometries. 
1  See  Chapter  VII,  §  5,  regarding  Formula  153  (the  same  as  353). 


236         THE  MAKING  OF  INDEX  NUMBERS 

crossing  may  be  the  crossing  of  two  time  antitheses  (53 
and  59,  or  54  and  60,  or  3  and  19,  or  4  and  20,  or  5  and  17, 
or  6  and  18),  or  the  crossing  of  two  factor  antitheses  (53 
and  54,  or  59  and  60,  or  3  and  4,  or  5  and  6,  or  17  and  18, 
or  19  and  20).  Thus,  it  merely  needs  to  conform  to  one 
test  in  order  to  conform  to  both.  This  can  be  said  of  no 
other  formula. 


Formulae  53 and  54  Applied  To  12  Leading  Crops 
(Prices) 

(After  WM.Persons) 


55 
54 


is* 


#?          &  90  '95  W  '05  10  75  '20 

CHART  47P.  Showing  almost  the  same  closeness  of  agreement  but  the 
presence  of  a  slight  bias,  53  always  exceeding  54,  except  in  the  one  year, 
1920.  The  common  origin  of  the  two  curves  is  1910. 

It  is  derivable  from  the  aggregatives  (53,  54,  59,  60), 
from  the  arithmetics  (3,  4, 5,  6),  or  from  the  harmonics  (17, 
18,  19,  20),  or  from  both  the  arithmetics  and  the  har- 
monics. Consequently,  unlike  other  formulae,  it  recurs 
in  its  various  r61es  again  and  again  (as  103,  104,  105,  106, 
153,  154,  203,  205,  217,  219,  253,  259,  303,  305),  being  en- 
countered, so  to  speak,  at  the  many  crossroads  in  our 
tables.  Its  constituent  formulae,  53  and  54,  are  likewise 
frequent  repeaters,  and  are  the  only  pair  of  formulae  which 
are  at  once  time  antitheses  and  factor  antitheses. 

Another  interesting  fact,  as  shown  in  the  Appendix,1 
*See  Appendix  I  (Note  to  Chapter  XIII,  §  9,  "  Proportionality  Test  ")• 


WHAT  IS  THE  BEST  INDEX  NUMBER?      237 

is  that,  while  Formula  353  is  a  perfectly  true  average, 
nine  of  its  twelve  rivals  (all  excepting  1353,  2353,  3353  — 
themselves  aggregatives)  are  not  true  averages.  They 
fulfill  the  definition  of  an  average  of  the  price  relatives 
only  in  case  the  quantity  relatives  are  all  equal. 

Another  peculiarity  is  that  the  aggregative,  alone  of 
all  index  numbers,  does  not  require  calculating  price 
ratios. 

Formulae  53 'and  54  Applied  To  12  Leading  Crops 
(Quantifier) 

( After  WMPersons) 


15* 


•80          V5  '90  '95  '00  '05  '10  75  *2Q 

CHART  47Q.     Analogous  to  Chart  47P. 

§  13.   Formulae  53  and  54  Already  in  Close  Agreement 

Last  but  not  least,  Formulae  53  and  54  are  also  in  actual 
fact  far  closer  together  than  any  other  of  the  primary 
formulae  which  are  crossed.  The  remarkable  closeness 
between  the  two  index  numbers  calculated  by  53  and  54 
is  not  an  accident  merely  happening  to  be  true  for  the  36 
commodities  here  selected. 

Professor  Persons  has  calculated  an  index  of  the  physi- 
cal volume  of  exports  for  1920  by  Formulae  53  and  54, 
obtaining  93.3  and  95.1  per  cent,  differing  by  two  per  cent. 

We  find  the  same  closeness  if  we  take  the  90  commodities 


238         THE  MAKING  OF  INDEX  NUMBERS 

("  materials  ")  for  which  Professor  Wesley  C.  Mitchell 
gives  the  data  in  the  report  of  the  War  Industries  Board.1 
These  are  given  in  Charts  45P  and  45Q  and  show  the  same 
closeness  of  Formulae  53  and  54  for  prices  and  so  also  the 
same  closeness  for  quantities.  What  is  equally  important 
to  note  is  that,  in  both  cases,  as  in  the  corresponding  case 

Formulae  55  and  54  Applied  To  12  Leading  Crops 

(Prices) 
(After  WdPersons) 


f}  W  7f  13  14  19  »  7/  n  19 

CHART  48P.     Analogous  to  Chart  47P. 

» 

of  36  commodities,  there  is  no  tendency  for  either  of  the 
two  curves  to  be  constantly  above  or  constantly  below 
the  other. 

Another  case  from  an  entirely  different  field  is  that  of 
the  prices  of  100  stocks  and  the  quantities  sold  on  the 
New  York  Stock  Exchange,  from  daily  quotations. 

1  Wesley  C.  Mitchell,  "History  of  Prices  during  the  War,  Summary" 
(War  Industries  Board,  Bulletin  No.  1),  p.  45.  Mitchell  works  out  only 
53  (for  prices  and  quantities)  but  as,  fortunately,  he  gives  the  data  for 
values,  it  is  easy  to  calculate  54.  Mitchell  uses  1913  as  100  per  cent  al- 
though the  real  base  of  calculation  is  1917.  Accordingly  in  the  charts  I 
have  used  1917  as  the  common  point.  The  corresponding  data  for  all 
the  1366  commodities  were  not  published  and,  although  a  search  was 
made  in  my  behalf,  they  cannot  be  found  even  in  manuscript.  Were 
this  possible  it  would  be  easy  to  calculate  53,  54,  353,  for  the  entire  1366. 


WHAT  IS  THE  BEST  INDEX  NUMBER?      239 

Here  again  the  closeness  of  Formulae  53  and  54  is  illus- 
trated, in  Charts  46P  and  46Q. 

Charts  47P,  47Q,  and  48P,  48Q  are  made  from  the 
figures  of  Professor  Persons  for  12  crops,  the  first  pair 
by  five  year  intervals,  and  the  second  pair  by  year  to 
year  intervals.1  In  these  cases  the  divergence  is  a 
little  greater  than  in  the  case  of  the  36  commodities.2 
Charts  48 P  and  48Q  also  give  the  chain  figures,  which 
show  a  considerable  deviation  from  the  fixed  base  figures. 

Formulae  53 'and  54  Applied  To  12  Leading  Crops 

(Quantities) 
(After  WWersons) 


]** 


»  //  »  13  14  19  16  77  78  19 

CHART  48Q.     Analogous  to  Chart  48P. 

In  all  these  crop  figures  there  is  discernible  the  effect 
of  an  inverse  correlation  between  the  price  and  quantity 
movements.  This  is  of  interest  to  the  student  of  index 
numbers  in  three  ways :  (1)  it  signifies  a  slight  modifica- 
tion of  the  proposition  that  Formulae  53  and  54  are  not 
subject  to  bias ;  (2)  it  confirms  the  proposition  that  any 
bias  in  the  fixed  base  system  is  intensified  in  the  chain 
system;  and  (3)  it  shows  that  such  a  bias  as  is  here 

1  Warren  M.  Persons,  "Fisher's  Formula  for  Index  Numbers,"  Review 
of  Economic  Statistics  (Statistical  Service  of  the  Harvard  University  Com- 
mittee on  Economic  Research,  Cambridge,  Mass.),  May,  1921,  pp.  103-113. 

2  Somewhat  greater,  even,  than  appears  at  first  glance,  as  the  scale  of 
these  four  charts  had  to  be  reduced  to  get  them  on  the  page.     The  little 
yardstick  in  the  charts  —  the  "5  %"  vertical  line  —  is  evidently  shorter 
than  that  in  all  preceding  charts,  indicating  that  in  this  chart  a  given  verti- 
cal distance  means  a  greater  per  cent  increase  than  in  former  charts. 


240         THE  MAKING  OF  INDEX  NUMBERS 

illustrated  —  a  sort  of  secondary  bias,  as  we  shall  see  — 
is  very  small.1 

§  14.  History  of  Formula  353 

The  constituent  Formulae  53  and  54  from  which  353 
is  constructed  are,  as  has  already  been  noted,  due  respec- 
tively to  Laspeyres  and  Paasche.  Formula  53,  or  Las- 
peyres,  is  the  most  practical  of  the  two  when  a  substitute 
for  353  has  to  be  used.  It  (53)  was  advocated  strongly 
and  ably  by  G.  H.  Knibbs,  Statistician  of  Australia.2 

Formula  53  being  identical  with  3,  it  has  been  used 
sometimes  as  an  arithmetic  average  with  base  year 
weighting  and  calculated  laboriously  as  such.  Apparently 
I  was  the  first  to  point  out  the  identity  of  the  two  formulae.3 
The  great  service  performed  by  Knibbs  was  to  point  out 
the  great  saving  in  time  in  calculating  53  as  an  aggregative 
rather  than  calculating  Formula  3  as  an  arithmetic  index 
number.  Knibbs  also  points  out  that  53  (and  the  same 
might  be  said,  though  less  emphatically,  of  the  other  ag- 
gregatives)  has  the  advantage  over  the  geometries  and 
other  types  of  being  easily  comprehended  by  the  gen- 
eral public.4  Formula  53  is  used  by  the  United  States 
Bureau  of  Labor  Statistics,  having  been  introduced  by 
Dr.  Royal  Meeker.  The  method  was  recently  endorsed 
in  a  resolution  (No.  81)  passed  by  the  British  Imperial 
Statistical  Conference  in  1920.  This  resolution  reads  : 


1  See  Appendix  I  (Note  to  Chapter  XI,  §  13). 

2  See  G.  H.  Knibbs,  "Price  Indexes,  Their  Nature  and  Limitations, 
the  Technique  of  Computing  Them,  and  Their  Application  in  Ascertaining 
the  Purchasing  Power  of  Money."  Commonwealth  Bureau  of  Census  and 
Statistics,  Labour  and  Industrial  Branch,  Report  No.  9,  McCarron,  Bird 
&  Co.,  Melbourne,  1918. 

8  Economic  Journal,  December,  1897,  pp.  517,  520.  See  also,  Pur- 
chasing Power  of  Money,  table  opposite  p.  418,  heading  of  Formulae  11  and 
12  and  discussion. 

4  Discussed  in  Chapter  XVI,  §  8. 


WHAT  IS  THE  BEST  INDEX  NUMBER?      241 

METHODS  OP  CONSTRUCTING  INDEX  NUMBERS 

That  the  index  numbers  should  be  so  constructed  that  their 
comparison  for  any  two  dates  should  express  the  proportion 
of  the  aggregate  expenditure  on  the  selected  list  of  representa- 
tive commodities,  in  the  quantities  selected  as  appropriate,  at 
the  one  date,  to  the  aggregate  expenditure  on  the  same  list  of 
commodities,  in  the  same  quantities,  at  the  other  date. 

This  phraseology  may  perhaps  be  taken  as  applicable, 
not  only  to  Formula  53,  but  also  to  54,  1153,  2153, 
3153,  4153.  Since  2153  is,  as  we  shall  see,  a  short  cut 
method  of  calculating  353,  we  may  practically  include 
353  also. 

So  far  as  I  know,  the  earliest  reference  to  the  formula 
here  numbered  353  is  that  made  by  C.  M.  Walsh  inciden- 
tally in  a  footnote  of  The  Measurement  of  General  Exchange 
Value  in  1901. 1  Walsh's  mention  of  it  had  escaped 
my  notice  until  he  called  my  attention  to  it  in  correspond- 
ing with  me  in  1920.  The  next  mention  appears  to  be  in 
my  Purchasing  Power  of  Money,  1911,  where  it  is  given 
as  Formula  16  in  the  table  opposite  page  418,  but  I  did 
not  then  appraise  it  as  the  best. 

Apparently  the  next  writer  to  mention  this  formula, 
and  with  high  approval,  is  Professor  A.  C.  Pigou  in  his 
Wealth  and  Welfare,  1912.2  He  regards  it  as  probably 
the  best  measure  for  comparing  price  levels  of  two  coun- 
tries. This  anticipation  by  Professor  Pigou  was  called 
to  the  attention  of  Mr.  Walsh  and  myself  by  Professor 
Frederick  R.  Macaulay. 

Next  in  order  comes  my  preliminary  paper  on  "  The 
Best  Form  of  Index  Numbers,"  read  December,  1920, 
where  I  advocated  353  as  the  best  or  "  ideal "  for- 

» p.  429. 

*  p.  46.  By  inadvertence  the  square  root  sign  is  omitted,  but  is  inserted 
in  the  later  book  Economics  of  Welfare,  1920,  p.  84. 


242         THE  MAKING  OF  INDEX  NUMBERS 

mula.1  Writing  contemporaneously,  without  knowl- 
edge of  my  work,  C.  M.  Walsh  added  this  formula  to  the 
other  index  numbers  recommended  by  him  as  "  perhaps 
the  best "  in  his  Problem  of  Estimation?  published 
February,  1921.  The  same  formula  was  reached  from  a 
still  different  angle  by  Professor  Allyn  A.  Young 3  as  the 
best  for  measuring  changes  of  the  general  price  level. 

Several  others  have  accepted  353  (e.g.  George  R. 
Davies,  Introduction  to  Economic  Statistics,  1922,  p.  86)  as 
best  for  certain  purposes.  It  is  a  great  satisfaction  to 
know  that  several  of  us  have  now  reached  the  conclusion 
that  this  formula  is  the  best,  even  if  some  still  add  safe- 
guarding qualifications.  I  think  we  may  be  confident 
that  the  end  is  being  reached  of  the  long  controversy 
over  the  proper  formula  for  an  index  number. 

Professor  Persons4  refers  to  the  index  number  which 
I  have  called  "  ideal "  as  "  Fisher's  Index  Number." 
This  was  doubtless  pursuant  to  the  too  generous  sugges- 
tion of  Mr.  Walsh  at  the  Atlantic  City  meeting.5  If  the 
conclusions  of  this  book  be  accepted,  I  think  my  proposed 
term  "  ideal  "  is  the  most  appropriate.  But,  if  my  name 
is  to  be  used,  Walsh's,  or  Walsh's  and  Pigou's  should  be 
used  also. 

1  Published,  with  discussion,  in  Quarterly  Publication  of  the  American 
Statistical  Association,  March,  1921. 

»  pp.  102-103. 

8  See  Quarterly  Journal  of  Economics,  "The  Measurement  of  Changes 
of  the  General  Price  Level,"  August,  1921,  p.  572. 

4  "Fisher's  Formula  for  Index  Numbers,"  Review  of  Economic  Statistics, 
May,  1921,  p.  103. 

6  See  Quarterly  Publication  of  the  American  Statistical  Association, 
March,  1921,  p.  544. 


CHAPTER  XII 

COMPARING  ALL  THE  INDEX  NUMBERS  WITH  THE 
"  IDEAL  "  (FORMULA  353) 

§  1.  All  Index  Numbers  Arranged  in  Order  of 
Their  Remoteness  from  Formula  353 

WE  have  chosen  Formula  353  as  the  most  nearly  ideal 
index  number,  have  measured  its  precision,  have  found 
that  the  12  others  in  our  list  which  have  the  best  inde- 
pendent claims  to  rival  Formula  353  coincide  with  it  for 
all  intents  and  purposes,  and  that  34  other  index  numbers, 
i.e.  those  free  merely  from  obvious  bias  or  freakishness, 
agree  with  it  nearly  enough  for  ordinary  requirements. 
And  now  we  can  look  back  and,  by  using  Formula  353 
as  a  standard  for  comparison  (or,  if  anyone  prefers,  any 
other  of  those  deserving  honorable  mention  in  our  contest), 
we  can  compare  all  the  other  133  formulae  with  that 
standard.  For  this  comparison  I  have  arranged  all  the 
formulae  in  order  of  then-  closeness  to  Formula  353.1 

Numerically,  Table  28  gives  all  the  1342  index  numbers 
in  the  order  of  remoteness  from  353,  beginning  with  the 
remotest  and  ending  with  353  itself.  The  figures  are  for 
prices,  not  quantities  (although  the  order  is  substantially 
the  same  hi  both  cases) ,  and  for  fixed  base  figures,  not  chain. 
In  each  case  the  formula  number  is  given  (in  the  first 
column)  for  identification.  Thus,  the  first  in  the  list  is 
Formula  12,  which  is  the  factor  antithesis  of  the  simple 
harmonic  index  number.  In  the  second  column  is  given 

1  For  the  method  used  see  Appendix  I  (Note  to  Chapter  XII,  §  1). 

2  These  form  only  119  different  ranks  because  those  tied  in  rank  are 
given  the  same  number.     Thus  the  second  number  in  the  list,  "118," 
applies  to  seven  different  index  numbers. 

243 


244 


THE  MAKING  OF  INDEX  NUMBERS 


the  letter  or  number  of  the  class,  out  of  the  seven  classes 
enumerated  in  Chapter  IX,  §  5,  to  which  the  index  num- 
ber belongs.  Thus  12  belongs  to  "  S,"  the  "  simple  "  group, 
being  a  derivative  of  the  simple  harmonic  index  number. 
In  order  to  simplify  the  picture,  the  list  of  134  is  sep- 
arated arbitrarily  into  several  classes  in  increasing  order 
of  merit.  The  first  twelve  index  numbers,  constituting 
the  first  of  these  classes,  are  labeled,  rather  harshly 
perhaps,  as  "  worthless  "  index  numbers  (to  designate 
the  fact  that  they  are  the  worst).  The  other  six  classes 
are  labeled  as  poor,  fair,  good,  very  good,  excellent,  and 
superlative.  Decimals  are  omitted  (as  superfluous  for 
the  comparisons)  from  all  classes  worse  in  rank  than  the 
"  very  good  "  and  these  are  given  but  one  decimal.  Only 
the  "  excellent "  and  "  superlatives  "  are  accorded  two  deci- 
mals. The  reader  will  quickly  form  a  mental  comparison 
of  various  formulae  by  running  his  eye  down  the  columns, 
especially  1917,  for  which  the  variations  are  the  greatest. 

TABLE  28.  INDEX  NUMBERS  BY  134  FORMULA  ARRANGED 
IN  THE  ORDER  OF  REMOTENESS  FROM  THE  IDEAL 
(353)  (AS  SHOWN  BY  THE  FIXED  BASE  FIGURES  FOR 
THE  PRICE  INDEXES) 

(1913  =  100) 


IDENTIFI- 

CLASS 

(INVERSE) 

CATION 

OP 

1914 

1915 

1916 

1917 

1918 

ORDER  OF 

NUMBER 

FORMULA 

MERIT 

WORTHLESS  INDEX  NUMBERS 


12 

8 

103 

101 

115 

172 

244 

119 

44 

M 

103 

106 

132 

196 

180 

118 

46 

M 

•« 

•• 

«• 

48 

M 

it 

•« 

" 

50 

M 

'• 

14 

it 

144 

M 

" 

M 

" 

146 

M 

" 

" 

•• 

1144 

M 

" 

" 

" 

42  =  142 

SM 

104 

108 

125 

167 

183 

117 

41=141 

8M 

98 

98 

108 

135 

190 

116 

1 

3 

96 

98 

124 

176 

187 

115 

51=151 

8 

96 

96 

108 

147 

173 

114 

COMPARING  ALL  THE  INDEX  NUMBERS    245 


TABLE  28  (Continued) 


IDENTIFI- 

CLASS 

(INVERSE) 

CATION 

OF 

1914 

1915 

191$ 

1917 

1918 

ORDER  OF 

NUMBER 

FORMULA 

MERIT 

POOR  INDEX  NUMBERS 


11 

8 

95 

96 

119 

158 

172 

113 

21=121 

S 

96 

97 

121 

167 

180 

112 

101 

S 

96 

97 

121 

167 

179 

111 

251  =351 

3 

97 

97 

111 

153 

169 

110 

102 

S 

102 

99 

113 

162 

208 

109 

243 

M 

102 

103 

119 

179 

174 

108 

245 

M 

•« 

44 

44 

44 

M 

M 

247 

M 

44 

44 

«• 

44 

44 

« 

249 

M 

•• 

M 

•• 

•« 

«• 

44 

343 

M 

44 

" 

M 

•« 

«• 

4fl 

345 

M 

«« 

•« 

•« 

«• 

" 

44 

1343 

M 

44 

" 

44 

44 

«« 

" 

5343 

M 

44 

" 

14 

•  •• 

44 

" 

211 

S 

99 

98 

117 

165 

205 

107 

9 

2 

101 

102 

118 

1  181 

187 

106 

52  =  152 

S 

97 

97 

115 

159 

165 

105 

7 

2 

101 

102 

118 

181 

187 

104 

14 

2 

102 

102 

117 

168 

190 

103 

15 

2 

100 

98 

111 

145 

167 

102 

13 

2 

99 

98 

111 

147 

169 

101 

301 

8 

99 

98 

117 

164 

193 

100 

8 

2 

09 

97 

111 

152 

167 

99 

10 

2 

99 

97 

111 

155 

169 

98 

16 

2 

101 

102 

117 

169 

189 

97 

241  =341 

SM 

101 

103 

116 

150 

186 

96 

22  =  122 

8 

102 

99 

113 

162 

194 

95 

31=131 

SM 

99 

99 

119 

164 

191 

94 

34 

M 

101 

105 

118 

166 

182 

93 

221  =321 

8 

99 

98 

117 

164 

187 

92 

33 

M 

100 

99 

107 

156 

169 

91 

43 

M 

101 

100 

108 

164 

168 

90 

45 

M 

" 

•« 

•• 

•• 

14 

47 

M 

«• 

•• 

•• 

44 

44 

49 

M 

44 

44 

44 

44 

14 

143 

M 

•• 

44 

M 

44 

" 

145 

M 

44 

14 

•• 

i« 

" 

1143 

M 

•f 

44 

44 

44 

" 

36 

M 

101 

104 

118 

165 

182 

89 

201 

S 

98 

97 

116 

164 

182 

88 

37 

M 

101 

100 

109 

164 

188 

87 

35 

M 

100 

99 

107 

160 

169 

86 

2 

S 

100 

96 

110 

153 

177 

85 

246 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  28  (Continued) 


IDENTIFI- 

CLASS 

(INVERSE) 

CATION 

OF 

1914 

1915 

1916 

1917 

1918 

ORDER  OF 

NUMBER 

FORMULA 

MERIT 

FAIR  INDEX  NUMBERS 


1134 

M 

101 

103 

118 

163 

182 

84 

1133 

M 

101 

100 

108 

163 

171 

83 

9051 

102 

103 

114 

160 

182 

82 

134 

M 

101 

103 

117 

163 

181 

81 

29 

1 

101 

101 

116 

170 

182 

80 

23 

1 

100 

99 

111 

154 

173 

79 

133 

M 

101 

100 

108 

160 

174 

78 

136 

M 

101 

103 

117 

162 

181 

77 

231  =  331 

SM 

100 

100 

117 

163 

187 

76 

1003 

1 

100 

101 

116 

171 

183 

75 

24 

1 

101 

101 

116 

165 

183 

74 

25 

1 

100 

99 

113 

152 

172 

73 

1013 

1 

100 

99 

113 

154 

173 

72 

27 

1 

100 

101 

116 

171 

182 

71 

38 

M 

101 

102 

117 

158 

180 

70 

1014 

1 

101 

101 

116 

165 

183 

69 

30 

1 

99 

98 

113 

159 

174 

68 

135 

M 

101 

100 

108 

162 

178 

67 

1004 

1 

99 

99 

113 

158 

173 

66 

39 

M 

101 

100 

109 

164 

178 

65 

28 

1 

100 

99 

113 

157 

172 

64 

6023  (*13-'14) 

100 

100 

112 

154 

173 

63 

32  =  132 

SM 

100 

102 

116 

162 

184 

62 

26 

1 

101 

101 

115 

165 

183 

61 

233 

M 

101 

102 

112 

161 

175 

60 

237 

M 

101 

101 

113 

161 

184 

59 

235 

M 

101 

102 

112 

163 

176 

58 

40 

M 

101 

102 

117 

160 

180 

57 

GOOD  INDEX  NUMBERS 


335 

M 

101 

101 

113 

162 

180 

56 

1333 

M 

101 

101 

113 

163 

176 

55 

5333 

M 

101 

101 

113 

162 

179 

54 

333 

M 

101 

101 

113 

161 

177 

53 

239 

M 

101 

101 

113 

162 

179 

52 

6023  ('13  &  '18) 

99 

99 

114 

160 

180 

51 

6023  ('13-'16) 

100 

100 

114 

157 

175 

50 

209 

0 

100 

100 

115 

167 

178 

49 

213 

0 

101 

100 

114 

157 

179 

48 

207 

0 

100 

100 

115 

166 

177 

47 

215 

0 

100 

100 

114 

156 

178 

46 

223 

0 

100 

100 

114 

159 

178 

45 

225 

0 

100 

100 

114 

159 

177 

44 

229 

0 

100 

100 

114 

164 

178 

43 

227 

0 

100 

100 

114 

164 

177 

42 

110 

0 

100 

100 

114 

162 

179 

41 

109 

0 

100 

100 

115 

163 

178 

40 

COMPARING  ALL  THE  INDEX  NUMBERS     247 


TABLE  28  (Continued} 


IDENTIFI- 

CLASS 

(INVERSE) 

CATION 

OF 

1914 

1915 

1916 

1917 

1918 

ORDER  OP 

NUMBER 

FORMULA 

MERIT 

VERY  GOOD  INDEX  NUMBERS 


1 

6053  ('13-'18) 

99.8 

99.9 

114.0 

161.6 

177.9 

39 

54* 

0 

100.3 

100.1 

114.4 

161.1 

177.4 

38 

108 

0 

100.2 

99.6 

114.0 

160.3 

177.9 

37 

53f 

0 

99.9 

99.7 

114.1 

162.1 

177.9 

36 

6053  ('13-'16) 

100.0 

100.0 

114.0 

161.9 

178.2 

35 

4153 

0 

100.1 

100.0 

114.4 

162.4 

178.3 

34 

309 

0 

100.2 

99.9 

114.3 

162.3 

178.4 

33 

107 

0 

100.1 

99.9 

114.4 

161.8 

176.6 

32 

4154 

0 

100.1 

99.9 

114.1 

161.2 

176.8 

31 

6053  ('13-'14) 

100.1 

100.1 

113.9 

161.3 

177.7 

30 

123 

0 

100.1 

99.9 

113.8 

162.1 

177.8 

29 

3153 

0 

100.2 

99.9 

114.2 

162.1 

176.9 

28 

307 

0 

100.1 

99.8 

114.2 

161.0 

177.3 

27 

*  54=4,  5,  18,  19,  59.  f  53=3,  6,  17,  20,  60. 

EXCELLENT  INDEX  NUMBERS 


323 

0 

100.13 

99.89 

113.99 

161.90 

177.98 

26 

124 

0 

100.16 

99.85 

114.25 

161.74 

178.16 

25 

3353 

0 

100.14 

99.90 

114.35 

161.94 

177.36 

24 

7053 

0 

100.09 

99.96 

114.03 

161.53 

177.90 

23 

126 

0 

100.12 

99.85 

114.20 

161.18 

177.36 

22 

325 

0 

100.12 

99.85 

114.19 

161.28 

177.35 

21 

1104 

0 

100.15 

99.84 

114.18 

161.58 

177.92 

20 

5307 

0 

100.15 

99.82 

114.21 

161.67 

177.84 

19 

1103 

0 

100.13 

99.91 

114.26 

161.93 

177.72 

18 

125 

0 

100.12 

99.87 

114.19 

161.37 

177.34 

17 

4353 

0 

100.13 

99.92 

114.26 

161.78 

177.52 

16 

3154 

0 

100.12 

99.92 

114.28 

161-77 

177.78 

15 

1303 

0 

100.14 

99.88 

114.22 

161.75 

177.82 

14 

1123 

0 

100.14 

99.89 

114.17 

161.62 

177.87 

13 

1124 

0 

100.12 

99.91 

114.28 

161.78 

177.73 

12 

SUPERLATIVE  INDEX  NUMBERS 


5323 

0 

100.13 

99.87 

114.09 

161.59 

177.67 

11 

1323 

0 

100.13 

99.90 

114.23 

161.70 

177.80 

10 

1153 

0 

100.13 

99.89 

114.20 

161.70 

177.83 

9 

1353 

0 

100.13 

99.89 

114.22 

161.71 

177.79 

8 

1154 

0 

100.12 

99.90 

114.24 

161.73 

177.76 

7 

2154 

0 

100.14 

99.90 

114.21 

161.69 

177.72 

6 

2353 

0 

100.13 

99.89 

114.22 

161.60 

177.67 

5 

2153 

0 

100.12 

99.89 

114.23 

161.52 

177.63 

4 

8054 

0 

100.12 

99.89 

114.21 

161.56 

177.65 

3 

8053 

0 

100.12 

99.89 

114.21 

161.56 

177.65 

2 

353* 

0 

100.12 

99.89 

114.21 

161.56 

177.65 

1 

*  353  =  103,  104,  105,  106,  153,  154,  203,  205,  217,  219,  253,  259,  303,  305. 


248         THE  MAKING  OF  INDEX  NUMBERS 

It  will  be  seen  that  of  the  12  formulae  which  we  found 
in  the  last  chapter,  each  on  its  own  independent  merits, 
to  be  the  closest  mates  to  the  ideal,  Formula  353,  two 
(307  and  309)  are  classed  as  "  very  good";  six  (323,  3353, 
325,  5307,  4353,  1303)  are  classed  as  "  excellent,"  and 
four  (5323,  1323,  1353,  2353)  are  classed  as  "superlative." 
That  is,  all  of  the  formulae  selected  as  best  on  independent 
grounds  also  prove  to  be  among  the  very  best  when  ranked 
on  the  basis  of  agreement  with  353. 

And  yet,  interspersed  with  these  12  are  others  just  as 
close  to  Formula  353,  though  not  exactly  fulfilling  both 
tests.  Most  of  these  are  the  various  combinations  of 
53  and  54.  These  two  formulae  are  so  extremely  close 
to  each  other  that  any  method  of  splitting  their  hair's 
difference  will  necessarily  agree  almost  absolutely  with 
353.  Thus  the  formula  closest  to  Formula  353  is  8053, 
the  arithmetic  average  of  53  and  54.  Although  8053 
does  not  fulfill  either  test,  it  comes  very  close  to  fulfilling 
both  and  to  coinciding  with  353  which  fulfills  both  ex- 
actly. All  of  the  other  "  superlative  "  index  numbers 
are  combinations  of  53  and  54. 

§  2.   Chart  Giving  Index  Numbers  in  Order 

Graphically,  we  can  get  a  much  quicker  and  clearer 
view  than  is  possible  by  mere  numerical  figures.  Chart  49 
gives  the  same  119  ranks  as  were  represented  in  Table  28. 
But  the  chart  includes,  in  addition,  the  chain  figures, 
represented,  in  the  usual  way,  by  small  balls.1  These 

1  The  distance  between  each  ball  and  the  curve  exhibits  the  disparity 
between  the  fixed  base  and  the  chain  figures.  This  distance  for,  say,  the 
year  1918  represents  the  net  cumulative  effect  of  the  disparities  of  all 
the  preceding  years.  In  order  to  show  how  much  disparity  there  has 
been  in  the  last  year  elapsed,  a  dark  vertical  line  is  inserted  (i.e.  extending 
from  the  1918  ball  to  the  point  where  that  ball  would  have  been  had  it 
remained  the  same  distance  from  the  curve  as  the  ball  of  the  last  year, 
1917) ;  and  likewise  for  each  other  year. 


COMPARING  ALL  THE  INDEX  NUMBERS    249 

RANKING  AS  TO  ACCURACY  ** 

OF 

ALL  INDEX  NUMBERS 

(I).  Worthless  Index  Numbers 
(Prices) 


73  '/+  '15  '16  77  7B 

CHART  49  (1).  These  index  numbers,  ranked  as  the  least  accurate,  include 
one,  the  simple  arithmetic  (1),  in  very  common  use,  and  another,  the  sim- 
ple aggregative  (51),  in  occasional  use.  The  six  index  numbers,  not  only 
disagree  widely  with  the  ideal  (353)  used  as  a  standard,  but  also  with  each 
other,  and  also  as  between  the  fixed  base  and  chain  figures  of  each  (as 
shown  by  the  balls  and  the  dark  verticals  —  the  displacement  of  each  ball 
from  the  curve  indicating  the  cumulative  divergence  of  the  chain  figures, 
and  the  dark  vertical  indicating  the  year-to-year  divergence). 


250         THE  MAKING  OF  INDEX  NUMBERS 


(2).  Poor  Index  Numbers 


Is* 


«r  74  '15  '16  77  'Id 

CHART  49  (2).  The  same  divergencies  in  less  degree  are  here  in  evidence, 
except  that  101  and  21  agree.  The  list  includes  two  which  have  been 
actually  suggested,  the  simple  harmonic  (11)  suggested  by  Coggeshall  and 
the  doubly  biased  arithmetic  (9)  suggested  by  Palgrave. 


COMPARING  ALL  THE  INDEX  NUMBERS    251 


(2)cont  Poor  Index  Numbers 
(Prices) 


CHART  49  (2,  continued).    This  set  includes  the  simple  median  (31). 


252         THE  MAKING  OF  INDEX  NUMBERS 


(5).  Fair  Index  Numbers 
(Prices) 


'13  74  75  '16  17  'Id 

CHART  49  (3) .  The  same  divergencies,  still  less  marked,  are  noted.  Of 
these  the  most  usable  is  9051,  a  quickly  calculated  rough-and-ready  index 
number. 

balls  and  the  dark  vertical  lines  attached  to  them  will  be 
more  especially  discussed  later.  At  present  they  may  be 
ignored  by  the  reader  so  that  his  attention  may  be  con- 
centrated on  the  ranking. 


COMPARING  ALL  THE  INDEX  NUMBERS    253 


(5)cont    Fair  Index  Numbers 
(Prices) 


7J  74  '15  '16  77  76 

CHART  49  (3,  continued).    This  list  includes  one  fcrm  of  Professor  Day's 
index  number  (6023). 

§  3.  The  Index  Numbers  Converge  toward 
Formula  353 

The  most  striking  fact  in  Table  28  and  Chart  49  is  the 
steady  natural  convergence  of  the  index  numbers  toward 


254 


THE  MAKING   OF  INDEX  NUMBERS 


W.  Good  Index  Numbers 

(Prices) 


'&  '1+  '/5  'If*  77  '/a 

CHART  49  (4).  The  disagreements  have  here  largely  disappeared,  whether 
as  between  each  curve  and  Formula  353,  or  as  among  themselves,  and  also 
as  between  the  fixed  base  and  chain  series.  This  list  includes  two  forms  of 
Professor  Day's  index  number  (6023). 

Formula  353.  This  would  not  be  true  if  we  had  arbi- 
trarily chosen  some  widely  different  curve  as  the  standard 
of  reference,  such  as,  say,  2  or  44.  It  is  noteworthy  that 
Formula  353  and  those  practically  coincident  with  it 
constitute  the  only  type  of  index  number  out  of  all  the 
numerous  varieties  which  can  boast  of  having  many  like 


(5)  Very  Good.  (6)  Excellent,  and 
(7) Superlative  Index  Numbers 
(Prices) 


(51  Very  Good  Ind.  Nos. 


<6).Excellent  Ind.  Nos 


(7).Sup*rlative  Ind.  Nos. 


yj  •/*  75  w  17  re 

CHART  49  (5, 6, 7).  All  the  divergencies  continue  to  disappear  until  they 
become  imperceptible.  The  "  very  goods  "  include  Laspeyres'  (53),  Paasche's 
(54),  and  Lehr's  (4153  and  4154).  The  "excellents "  include  one  of  Walsh's 
(1123),  and  Lehr's  rectified  by  Test  2  (4353).  The  "superlatives"  include 
the  above  Walsh's  rectified  by  Test  2  (1323),  two  of  Walsh's  (1153  and 
1154),  the  same  rectified  by  Test  2  (1353),  Edgeworth's  and  Marshall's 
(2153),  another  of  Walsh's  (2154),  the  rectification  by  Test  2  of  the  two  lat- 
ter (2353),  Drobisch's  (8053),  and  the  "  ideal"  353,  used  as  standard  for 
all  of  the  Charts  49. 


256         THE  MAKING  OF  INDEX  NUMBERS 

it.  Thus,  if  any  one  should  contend  that  Formula  2 
was  the  best  index  number  and  should  try  to  arrange  the 
formulae  in  the  order  of  closeness  to  2,  he  would  not  find 
the  picture  altogether  unlike  that  now  before  us.  No.  2 
would  stand  very  much  alone,  its  closest  neighbors  all 
being  distant  from  it.  Furthermore,  the  index  numbers 
which  we  have  chosen  as  the  best  would,  in  such  an 
arrangement,  though  no  longer  placed  at  the  culminating 
end  of  the  list,  still  keep  close  together.  As  the  list  now 
stands,  almost  no  index  numbers,  far  away  from  353  but 
neighbors  to  each  other,  are  close  enough  neighbors  to  have 
any  strong  family  resemblance.  There  is  one  exception ; 
namely,  the  pair  of  101  and  21,  which  have  already  been 
noted  as  the  best  in  the  hierarchy  of  simple  index  numbers. 

Again,  the  119  varieties  in  our  chart  vary  about  equally 
on  the  opposite  sides  of  Formula  353,  even  though  the 
modes  and  medians  are  included,  as  is  shown  by  the  follow- 
ing averages  of  Table  29.  * 

Except  for  the  "  worthless  "  class  each  class  averages 
very  close  to  353,  showing  that  the  variations  above  and 
below  are  about  equal,  as  was  to  be  expected.  What  has 
been  said  would  still  be  true  even  if  we  should  leave  out 
of  consideration  so  many  types  of  averaging  53  and  54. 
In  short,  Formula  353  (or  any  equivalent)  is  the  evident 
goal  of  the  complete  set  toward  which,  as  toward  no 
1  other,  they  tend  to  converge. 

§  4.  Many  Besides  Formula  353  Pass  Muster 

It  will  be  seen  that  by  pronouncing  Formula  353  to  be 
the  best  index  number,  it  is  not  implied  that  it  is  separated 
by  a  wide  gulf  from  all  others.  On  the  contrary,  one  of 

1  Strictly  the  geometric  average  should  be  used ;  but,  except  for  the  first 
few  index  numbers  (where  it  was  used),  this  would  not  differ  appreciably 
from  the  arithmetic,  which,  for  ease  of  calculation,  was  used  in  all  other 


COMPARING   ALL  THE   INDEX  NUMBERS    257 


TABLE  29.    AVERAGES  OF  EACH  OF  THE  VARIOUS  CLASSES 
OF  INDEX  NUMBERS 


CLASSES 

1914 

1915 

1916 

1917 

1918 

Worthless 

100. 

101. 

118. 

164. 

193. 

Poor 

96. 

99. 

115. 

162. 

181. 

Fair 

101. 

101. 

114. 

161. 

179. 

Good 

100. 

100. 

114. 

161. 

178. 

Very  good 

100.1 

99.9 

114.1 

161.6 

177.6 

Excellent 

100.13 

99.88 

114.20 

161.65 

177.71 

Superlative 

100.13 

99.89 

114.21 

161.64 

177.72 

Average    of    all 
classes 

99.35 

100.06 

114.37 

161.83 

179.46 

353 

100.12 

99.89 

114.21 

161.56 

177.65 

our  main  conclusions  is  that  there  are  others  which  are 
really  just  as  accurate.  It  is  only  by  literally  splitting 
hairs  that  we  can  claim  any  superiority  in  accuracy  of  353 
over  its  fellow  "  superlatives,"  and  then  only  with  doubt. 
There  is  less  room  for  doubt  as  to  the  superiority  of  353 
over  the  "excellent"  index  numbers,  but  the  degree  of  its 
superiority  is  negligible.  In  fact,  judged  by  ordinary 
practical  standards,  we  can  extend  the  equality  to  the 
"  very  good  "  or  even  to  the  "  good." 

To  put  these  comparisons  in  figures,  let  us  take  1917 
in  which  the  variations  are  almost  always  the  greatest. 
Among  the  "  superlative  "  the  smallest  index  number  is 
161.52,  and  the  largest  161.73,  while  the  "  ideal,"  Formula 
353,  is  161.56.  Among  the  "  excellent  "  the  smallest  is 
161.18  and  the  largest  is  161.94.  Among  the  "very 
good "  the  smallest  and  largest  are  160.3  and  162.4. 
Among  the  "  good,"  they  are  156  and  167 ;  among  the 


258         THE  MAKING  OF  INDEX  NUMBERS 

"  fair,"  152  and  171 ;  among  the  "  poor,"  145  and  181 ; 
and  among  the  "  worthless,"  135  and  196. 

In  percentages  these  figures  show  the  maximum  de- 
viation from  the  ideal  (161.56)  to  be  as  follows  :  among  the 
"  superlative,"  .1  per  cent ;  among  the  "  excellent,"  .2  per 
cent ;  among  the  "  very  good,"  .8  per  cent ;  among  the 
"  good,"  3.7  per  cent ;  among  the  "  fair,"  6.2  per  cent ; 
among  the  "poor,"  11.7  per  cent;  among  the  u worth- 
less," 21  per  cent. 

How  far  can  we  go  in  letting  the  less  accurate  index 
numbers  pass  muster  as  good  enough  ?  The  answer  will 
vary,  of  course,  according  to  the  standards  we  set  in  any 
particular  case.  In  practice,  it  is  seldom  that  our  stand- 
ards require  a  closer  approximation  than  two  per  cent. 
On  this  basis  we  may  admit  as  usable  index  numbers  all 
of  the  11  "  superlative,"  the  15  "  excellent,"  the  11  "  very 
good,"  and  most  of  the  16  "  good,"  or  nearly  40  per  cent 
of  the  134  index  numbers  in  all.  These  are  all  in  the  "  0  " 
or  middle  tine  class,  i.e.  they  include  all  except  the  biased 
and  the  freakish  index  numbers,  which  is  in  accordance 
with  the  findings  of  the  previous  chapter. 

§  5.   Comments  on  Modes,  Medians,  and  Simples 

A  glance  at  the  class  symbols  in  the  second  column  of 
Table  28  shows  that  "  S  "  (or  the  simples  and  their 
derivatives)  and  "  M  "  (the  modes  and  medians)  are  mostly 
far  away  from  Formula  353 ;  that  the  "  2's  "  are  next 
farthest  from  353,  the  "  1's  "  next,  and  the  "O's  "  last. 

The  rankings  of  the  simples  are  of  interest.  When 
we  were  comparing  the  simples  among  themselves,  we  con- 
demned the  arithmetic  and  harmonic  and  their  antitheses 
(Formulas  1,  2,  11,  12)  on  the  sole  ground  of  bias;  we 
did  not  condemn  them  on  the  ground  of  freakishness  of 
weighting,  for  then  the  simple,  or  even,  weighting  was 


COMPARING   ALL  THE  INDEX  NUMBERS     259 

assumed  to  be  correct.  But  now  that  we  are  applying 
higher  standards  and  comparing  the  simples  themselves 
with  the  best  weighted  index  number  (Formula  353),  we 
condemn  every  simple  formula,  even  Formula  21,  on  the 
ground  of  freakish  weighting,  while  still  condemning 
Formulae  1,  2, 11, 12  on  the  further  ground  of  bias.  These 
latter  formulae  are  thus  doubly  bad,  combining  both 
freakishness  and  bias,  despite  the  fact  that,  in  some  cases, 
the  two  happen  to  neutralize  each  other.  Thus  Formula 
2  for  prices  happens  to  agree  closely  with  353  for  1914  and 
1918,  but  not  for  other  years.  Consequently  these  four 
formulae  stand  at,  or  near,  the  extreme  top  of  Table  28. 

It  should  also  be  noted  that  the  modes  in  particular 
occur  in  clusters  in  almost  random  order,  and  not  in  the 
order  of  their  conformity  to  tests.  Normally,  as  shown 
in  the  cases  of  the  other  varieties,  the  "  rectification  " 
process  actually  rectifies.  For  instance,  in  Table  28, 
we  find  that  the  primary  and  biased  weighted  geometries, 
Formulae  23,  24,  25,  26,  27,  28,  29,  30,  precede  the  (singly) 
rectified  Formulae  123,  1123,  124,  1124,  125,  126,  223, 
225,  227,  229,  and  these,  in  turn,  precede  the  doubly 
rectified  323,  325,  1323,  5323  (except  that  323  and  325 
are  slightly  out  of  the  prescribed  order).  Moreover,  all 
these  several  geometries  have  each  a  separate  rank  in  the 
list,  whereas  all  of  the  25  modes  (including  even  the 
simples)  occur  in  a  few  clusters  and  with  almost  no  regard 
to  any  systematic  order.  For  instance,  we  find  the  un- 
rectified  modes,  Formulae  44,  46,  48,  50,  not  preceding  the 
rectified  Formulae  144,  146,  1144,  but  clustered  in  exactly 
the  same  rank  with  them  and  with  one  another.  Again, 
we  find  243,  245,  247,  249,  343,  345,  1343,  5343  likewise 
clustered  at  identically  the  same  rank.  And  the  last- 
named  cluster  (consisting  of  rectified  index  numbers) 
precedes,  instead  of  follows  as  it  should,  the  cluster,  43, 


260         THE  MAKING  OF  INDEX  NUMBERS 

45,  47,  49,  143,  145,  1143  (comprising  mostly  unrectified 
index  numbers) .  The  medians  behave  considerably  better 
but  they  also  are  immobile  as  contrasted  with  others. 

The  table  completes  the  evidence  that  what  makes  a 
bad  index  number  is  either  freakishness  or  bias,  and  that 
the  bias  can  be  thoroughly  eliminated  by  the  rectifica- 
tion process,  while  freakishness  cannot.  Barring  index 
numbers  subject  to  these  defects,  all  index  numbers  are 
good.  In  other  words,  all  in  group  "0,"  which  lie  in 
the  middle  tine  of  the  five-tined  fork,  are  good.  With 
few  exceptions,  every  good  index  number  obeys  at  least 
one  of  the  two  tests.  The  exceptions  are  Formulae  53,  54, 
6023,  6053,  7053,  8053,  8054,  all  of  which,  while  they  fail 
to  obey  either  test,  come  very  near  to  obeying  both  tests. 

§  6.  The  Simple  Median  Nearer  the  Ideal  than 
the  Simple  Geometric 

We  are  now  ready  to  return  to  the  unfinished  discussion 
of  the  median.  It  is  one  of  the  interesting  and  surprising 
results  of  the  comparisons  in  the  complete  list  of  formulae 
that  the  simple  median  has  a  better  rating  than  any  other 
simple  index  number.  The  order  of  increasing  merit  of 
the  simples  as  here  shown  is:  Formulae  41  (worthless), 
1  (worthless),  51  (worthless),  11  (poor),  21  (poor),  31 
(poor).  The  median  thus  not  only  outranks  the  mode, 
which  was  to  be  expected,  and  the  simple  arithmetic,  so 
much  in  vogue,  but  even  the  geometric.  After  what  has 
been  said  as  to  the  freakishness  of  the  median  and  the 
virtues  of  the  geometric,  it  might  have  been  expected  that 
the  median  would  rank  among  the  worst  of  the  simples, 
and  that  Formula  21  would  rank  as  the  best.  And,  as 
we  have  seen,  when  we  assume  that  simple  or  equal  weighting 
is  the  right  weighting,  the  order  of  merit  would  make 
Formula  21  best  and  31  far  inferior.  But,  of  course, 


COMPARING  ALL  THE  INDEX  NUMBERS    261 


simple  weighting  never  really  is  the  right  weighting,  and 
our  table  of  merit  is  based  not  on  simple  but  on  true 
weighting.  On  such  a  scale,  Formula  31  seems  to  outrank 
21. 

The  comparison  of  Formulae  21  and  31  as  to  nearness 
to  353  may  be  presented  numerically  as  follows : 

TABLE  30.  ACCURACY  OF  SIMPLE  GEOMETRIC  AND  SIMPLE 
MEDIAN,  JUDGED  BY  THE  STANDARD  OF  FORMULA  353 
FOR  36  COMMODITIES 

(Prices) 


FORMULA  No. 

1913 

1914 

1915 

1916 

1917 

1918 

21 

100 

96 

97 

121 

167 

180 

31 

100 

99 

99 

119 

164 

191 

353 

100 

100 

100 

114 

162 

178 

Evidently,  the  median  (31)  is  somewhat  nearer  the  ideal  (353)  than  is 
the  geometric  (21)  in  1914,  1915,  1916,  and  1917,  —  by  three  per  cent, 
two  per  cent,  two  per  cent,  and  two  per  cent  respectively.  It  is  farther 
away  only  in  1918,  —  by  six  per  cent. 

We  may  further  test  the  conclusion  reached  by  comparing  Formulae 
21  and  31  as  applied  to  quantities.  The  figures  follow : 

TABLE  31.  ACCURACY  OF  SIMPLE  GEOMETRIC  AND  SIMPLE 
MEDIAN,  JUDGED  BY  THE  STANDARD  OF  FORMULA  353 
FOR  36  COMMODITIES 

(Quantities) 


FORMULA  No. 

1913 

1914 

1915 

1916 

1917 

1918 

21 

100 

98 

111 

121 

119 

115 

31 

100 

99 

107 

117 

119 

121 

353 

100 

99 

109 

119 

119 

125 

Here  again  Formula  31  shows  to  better  advantage  than  21,  being  one 
per  cent  superior  in  1914  and  five  per  cent  superior  in  1918,  and  scoring 
a  tie  in  1916,  1917,  and  1915.  In  1918  one  unimportant  commodity, 
skins,  the  quantity  of  which  fell  enormously  while  most  others  rose, 
spoils  the  sensitive  geometric,  —  even  making  the  average  movement 
seem  to  be  downward  when  it  is  really  upward,  —  but  it  has  no  influence 
on  the  insensitive  median. 


262 


THE  MAKING   OF  INDEX  NUMBERS 


Further  confirmation  of  our  conclusion  is  found  by  a  study  of  the  1437 
commodities  used  by  the  War  Industries  Board.  The  weighted  aggregative 
(Formula  53)  was  the  index  number  employed  there,  and  may  here  be 
used  as  our  standard  in  lieu  of  353.  I  have  computed  the  simple  geometric 
and  median.  The  results,  which  are  per  prices,  are  as  follows : 

TABLE  32.  ACCURACY  OF  SIMPLE  GEOMETRIC  AND  SIMPLE 
MEDIAN,  JUDGED  BY  THE  STANDARD  OF  FORMULA  53 
FOR  1437  COMMODITIES 

(Pre-war  year  July,  1913 -July,  1914=100  per  cent) 


FORMULA  No. 

1913 

1914 

1915 

1916 

1917 

1918 

21 

101 

101 

108 

138 

174 

198 

31 

101 

100 

101 

122 

162 

196 

53 

101 

99 

102 

126 

175 

194 

We  note  that  Formula  31  is  as  close  to  53  as  is  21  for  1913  and  closer 
in  1914  by  one  per  cent,  in  1915  by  five  per  cent,  1916  by  six  per  cent, 
and  1918  by  one  per  cent.  The  only  case  to  the  contrary  is  1917,  where 
Formula  21  is  the  closer  by  seven  per  cent.  Thus  Formula  31  is  superior 
to  21  for  the  prices  of  1437  commodities,  just  as  it  was  for  the  prices  and 
quantities  of  the  36.  Unfortunately  we  lack  the  data  for  calculating  the 
quantity  ratios  of  the  1437  commodities. 

Chart  50  shows  the  relations  of  the  simple  median  and  simple  geometric 
for  1917  for  commodities  taken  from  our  list  of  36  by  lot,  beginning  with 
three  and  including  every  odd  number.  It  will  be  seen  that,  as  compared 
with  353  (the  dotted  line)  it  is  nip  and  tuck  between  21  and  31  as  to  their 
closeness  to  353,  except  where  the  number  of  commodities  falls  below  11, 
when  21  is  decidedly  the  better.  In  spite  of  its  insensitiveness,  as  shown 
by  its  few  changes,  No.  31  averages  as  close  to  353  as  does  21. 

Is  this  apparent  superiority  of  Formula  31  an  accident?  It  is  hard 
to  say,  but  I  am  inclined  to  think  that,  at  any  rate,  21  is  not  on  the  average 
superior  to  31. 

Professor  Edgeworth  has  advocated  the  simple  median  on  the  ground 
that  it  cannot  be  so  easily  influenced  by  extreme  aberrations  of  one  or  two 
individual  commodities  of  small  importance.  In  simple  (or  equal)  weight- 
ing, commodities  of  small  importance  are  in  some  cases  endowed  with 
undue  influence.  This  is  so  in  the  case  of  the  geometric  (or,  for  that  matter 
also  the  arithmetic  and  harmonic),  whereas  in  the  case  of  the  median  such 
extreme  variations  produce  no  disturbance  whatever.  This  argument 
of  Edgeworth's  is  sound,  although  at  first  sight  it  seems  to  conflict  with 
some  of  the  lines  of  reasoning  heretofore  used  in  this  book.  While  Pro- 
fessor Edgeworth  praises  the  median  because  it  is  not  exaggeratedly  sen- 
sitive, it  would  seem  that  I  have  condemned  it  just  because  it  is  not  a 
sufficiently  sensitive  barometer.  This  conflict  of  opinion,  however,  is 
more  apparent  than  real.  Insensitiveness  is  an  unmitigated  evil  in  a 


COMPARING  ALL  THE  INDEX  NUMBERS    263 

carefully  weighted  index  number,  for  it  prevents  some  of  the  commodities 
from  having  their  proper  influence.  This  is  likewise  true  also  of  a  simple 
index  number  provided  its  (even)  weighting  happens  not  to  be  too  far  from 
the  true  weighting.  But  when  this  even  weighting  happens  to  differ  enor- 
mously from  the  true  weighting,  as  is  frequently  —  probably  usually 
— the  case,  the  matter  is  not  so  easily  disposed  of.  In  that  case  it  may  well 
be  that  the  insensitiveness  of  the  median  by  preventing  an  undue  influence 
of  extreme  aberrations  of  unimportant  commodities  may  more  than  make 
up  for  any  delinquency  in  preventing  the  due  influence  of  important  com- 
modities. Such  a  net  benefit  is  pretty  certain  to  accrue  when  the  unim- 
portant commodities  are  the  most  extreme  in  their  aberrations,  and  the 
important  ones,  the  least.  And,  possibly,  this  is  what  we  usually  find.  It 
follows  that  when  we  are  forced  to  use  a  simple  index  number  as  a  make- 

Simple  Geometric  and  Simple  Median,  Compared  with  Ideal 
\  for  Different  Numbers  of  Commodities 


•\^ ^  353 

\5% 

35     7     9     II     13    15     17    19    21     23   25    27  29    31    33   35 

NUMBER  OF  COMMODITIES 

CHART  50.  Showing  that  the  simple  median  (31)  and  simple  geometric 
(21)  are,  on  the  whole,  about  equally  near  the  ideal  (353)  for  the  seventeen 
different  numbers  of  commodities  except  in  the  cases  of  3,  5,  7,  and  9  com- 
modities, when  the  geometric  is  distinctly  nearer. 

shift  for  a  carefully  weighted  one  because  we  lack  the  data  for  weighting, 
the  simple  median  may  as  well  masquerade  as  a  weighted  index  number 
as  the  simple  geometric.  The  median  cannot  go  far  wrong  when  the 
really  important  commodities  do  not  disperse  very  widely,  whereas  the 
geometric  is  apt  to  be  thrown  out  of  this  middle  course  by  giving  a  vastly 
exaggerated  influence  to  a  few  unimportant  but  widely  aberrant  com- 
modities. 

Whether  the  important  commodities  really  do  usually  keep  near  the 
middle  of  the  road  as  compared  to  the  unimportant  is  doubtful,  however. 
Of  the  36  commodities  whose  prices  changed  from  1913  to  1914,  the  middle 
18  price  relatives  were  much  less  important  than  the  other  18,  i.e.  than  the 
9  whose  price  relatives  were  the  highest  and  the  other  9  whose  price  relatives 
were  the  lowest.  The  relative  unimportance  of  the  middle  18  is  best  meas- 
ured by  their  total  weights  (taken,  say,  as  the  mean  between  the  1913 
and  1914  values).  This  total  in  1914  was  equal  to  only  3692  out  of  a 
total  for  all  36  commodities  of  13024,  or  considerably  less  than  half.  In 


264         THE  MAKING  OF  INDEX  NUMBERS 

other  words,  the  average  commodity  near  the  middle  of  the  price  movement 
was  less  important  than  the  average  commodity  near  the  extremes  of 
movement.  This  is  true  in  all  the  years.  In  1915,  the  middle  18  commod- 
ities had  weights  of  4062  out  of  a  total  of  13588 ;  in  1916,  4746  out  of  15157  ; 
in  1917,  5776  out  of  17857;  in  1918,  6086  out  of  19307,  —  in  all  cases  less 
than  half.  Nevertheless,  in  spite  of  these  facts  for  the  36  commodities, 
the  simple  median,  as  we  have  seen,  is  slightly  nearer  the  ideal  than  is 
the  simple  geometric. 

Our  conclusion  is  that  the  simple  median,  except  when 
there  are  very  few  commodities,  is  probably  at  least  as 
good  on  the  average  as  a  substitute  for  a  weighted  index 
number  as  is  the  simple  geometric. 

Precisely  the  same  arguments  for  and  against  the  simple  median  (com- 
pared with  the  simple  geometric)  apply  also  to  the  simple  mode.  But 
in  this  case  the  balance  is  certainly  against  the  mode,  the  mode  being  far 
more  freakish  than  the  median.  The  mode,  Formula  41,  is  further  from  353 
than  is  21  in  Chart  49.  This  is  for  prices.  The  same  is  true  as  to  quantities. 
The  same  is  also  true  for  the  1437  commodities,  the  simple  mode  (41) 
being  99,  99,  99,  108,  145,  173,  as  against  the  geometric  (21)  which  runs 
101, 101, 108, 138, 174,  198,  while  the  aggregative  (53)  used  as  the  standard 
by  which  to  judge  between  21  and  31,  runs  101,  99,  102,  126,  175,  194. 

§  7.   Slight  Revision  of  the  Order  of  the  Best  Formulae 

The  order  of  merit,  which  we  have  found,  was  deter- 
mined quite  mechanically  and  doubtless  this  order, 
toward  the  end  where  the  competition  for  first  place  is  so 
close,  is  somewhat  accidental  and  would  vary  considerably 
if  calculations  were  made  with  other  data.  The  last  score 
or  two  of  formulae  are  practically  all  alike  in  accuracy. 
If  we  are  to  discriminate  at  all  among  these  it  is  better 
not  to  be  guided  wholly  by  mechanical  methods.  We 
may  revise  slightly  the  order  of  precedence.  It  is  doubtless 
an  accident  that  places  2153  one  place  nearer  the  ideal 
than  2353  which,  on  independent  grounds,  should  be  the 
better  formula.  Doubtless  ordinarily  it  is  the  closer  to 
353  and  is  actually  found  to  be  so  in  other  cases. 

Without  arguing  all  the  fine  distinctions  which  might  be 
drawn,  I  shall,  somewhat  dogmatically,  pronounce  my 


COMPARING  ALL  THE   INDEX  NUMBERS    265 

own  final  judgment  as  to  the  true  order  of  precedence  in 
accuracy,  taking  into  account  all  the  considerations  in 
this  and  the  preceding  chapter.  This  (increasing)  order 
of  merit  is :  309,  307,  5307,  1303,  4154,  4153,  3154,  3153, 
4353,  3353,  1124,  1123,  124,  126,  123,  125,  1154,  1153, 
2154,  2153,  323,  325,  8054,  8053,  1323,  1353,  5323, 
2353,  353.  These  I  should  call  the  29  best  formulae 
with  only  infinitesimal  preferences  among  them.  The 
list  has  been  intended  to  include  all  formulae  satisfying 
both  tests  (barring  medians,  modes,  simples,  and  their 
derivatives).  It  will  be  noticed,  however,  that  the  list 
includes  a  number  of  formulae  obeying  only  one  test  and 
two  (8053  and  8054),  very  excellent  ones,  obeying  neither. 

This  list  contains  none  of  the  formulae  in  common  use, 
most  of  which  are  objectionable  because  of  bias  or  freakish- 
ness.  This  sheaf  of  29  accurate  formulae  represents  the 
best  of  the  large  crop  reaped  from  the  seed  of  the  46 
primary  formulae.  The  29  are  all  within  less  than  one-half 
of  one  per  cent  of  the  "  ideal,"  353.  So  far  as  accuracy 
is  concerned  any  one  of  them  is  good  enough  to  serve 
for  all  practical  purposes.  Moreover,  none  outside  of 
this  list  need  ever  be  used  for  any  purpose  where  great 
accuracy  is  demanded,  although  about  as  many  other 
formulae  are  accurate  enough  for  most  purposes.  As 
to  other  considerations  than  accuracy,  more  will  be  said 
later. 

Few  writers  besides  Walsh  have  tried  to  go  outside 
what  are  here  called  the  primary  formulae.  The  usual 
attitude  is  to  observe  regretfully  that  "  different  ways  of 
computing  index  numbers  lead  to  different  results," 
and  then  either  to  shrug  the  shoulders  in  despair  of  any- 
thing better,  as  much  as  to  say  "  you  pays  your  money  and 
you  takes  your  choice/'  or  vaguely  to  contend  that  "some 
kinds  of  index  numbers  are  good  for  some  purposes  and 


266          THE  MAKING   OF  INDEX  NUMBERS 

some  for  others."  In  view  of  what  we  have  found  as  to 
bias,  rectification,  and  the  close  agreements  in  the  results, 
I  do  not  see  how  any  reasonable  man  can  henceforth 
continue  to  take  either  of  these  views. 

§  8.   Conclusions 

What,  then,  are  the  results  of  the  comparisons  among 
the  134  varieties  of  index  numbers?  The  chief  results 
seem  to  be : 

1.  The  only  really  unreliable  class  of  formulae  are  those 
which  are  distinctly  freakish,  whether  because  of  a  freak- 
ish type,  as  in  the  case  of  the  modes  and,  in  less  degree 
the  medians,  or  on  account  of  a  freakish  weighting,  as  in 
the  case  of  the  simples. 

2.  Formulae  which  are  merely  biased  can  always  be 
thoroughly  rectified  by  mating  with  formulae  of  equal, 
but  opposite,  bias. 

3.  Consequently,  in  Table  28,  all  the  biased  formulae 
(unless  of  freakish  origin)  take  their  places  with  great 
regularity  of  order ;  first,  the  doubly  biased  (the  "  2  +  " 
and  "  2  —  "  classes  occurring  side  by  side),  and  then  the 
singly  biased  in  the  same  way. 

4.  Any  type  of  formula,  with  the  single  exception  of 
the  incorrigible  mode  (which  in  our  Table  28  never  scores 
better  than  "  poor  "),  can,  by  passing  through  our  two 
rolling  mills  of  rectification  (Test  1  and  Test  2),  be  straight- 
ened out  into  a  good  index  number.     All  the  roads  lead 
to  Rome,  —  whether  the  roads  be  the  arithmetic,  the 
harmonic,  the  geometric,  or  the  aggregative. 

5.  Even  the  median,  which  is  fairly  freakish  by  nature, 
turns  out  in  the  end,  when  doubly  rectified,  to  be  at  least 
"good"    (viz.    Formulae   335,    1333,    5333,   333,  —  also 
239,  although  only  once  rectified).     Probably,  if  a  large 
number  (instead  of  only  36)  of  commodities  were  taken, 


COMPARING  ALL  THE  INDEX  NUMBERS    267 

the   median   would    come    considerably    closer    to    the 
"  ideal." 

6.  As  to  the  mode  also,  some  improvement  may  be 
expected  by  increasing  the  number  of  commodities.     Un- 
fortunately, we  lack  the  data  for  testing  weighted  modes, 
and  their  rectifications,  for  a  large  number  of  commodities. 
Judging  from  such  indications  as  are  at  hand,  I  venture 
the  guess  that,  for  100  or  200  commodities,  the  rectified 
weighted  mode  would  agree  with  the  ideal  within,  say, 
two  or  three  per  cent. 

7.  Just  as  any  type  of  index  number  (with  the  possible 
exception  of  the  freakish  mode)  can  be  rectified  to  agree 
approximately  with  the  ideal,  so  any  system  of  weighting,1 
excepting  such  freakish  weighting  as  the  "  simple, "  can 
be  rectified.     It  matters  not  whether  an  index  number, 
to  start  with,  be  weighted  according  to  systems  I, 1 1,  III, 
IV,  or  any  crosses  between  them.    After  rectification  by 
both  tests  the  resulting  index  number  will  invariably 
emerge  (except  for  modes)  as  competent.    In  fact,  in 
one  case,  even  simple  weighting  turns  out  fairly   well. 
The  simple  median,  a'fter  twofold  rectification,  becomes 
a  "  fair  "  index  number. 

8.  Every  doubly  rectified  index  number  (excepting  the 
modes  and  simples)  is  at  least "  good."   Four  (medians)  are 
classed  as  "  good  "  ;  two  (arithmetic-harmonic)  are  classed 
as  "  very  good  " ;  six  (arithmetic-harmonic,  geometric, 
and  aggregative)  are  classed  as  "  excellent "  ;  and  five  (ge- 
ometric and  aggregative)  are  classed  as  "  superlative." 

9.  Some  53  index  numbers  will  pass  muster  as  at  least 
"  good,"  of  which  the  five  worst  are  medians  and  the  11 
best  are  aggregatives  and  geometries  (the  "  superlative  "). 

1  Because  so  much  importance  has  hitherto  been  attached  to  the  prob- 
lem of  weighting,  I  have  included  an  Appendix  (II)  on  "The  Influence  of 
Weighting."  But  it  is  not  essential  to  the  course  of  the  argument  of  this 
book. 


268         THE  MAKING  OF  INDEX  NUMBERS 

All  the  intervening  37  index  numbers  are  aggregatives, 
geometries,  and  arithmetic-harmonics  (unless  we  call 
Formulae  207  and  209  arithmetics  alone,  and  213  and 
215  harmonics  alone). 

10.  Consequently,    the   nature    of   the   index    number 
formula  (whether  arithmetic,  harmonic,  geometric,  median, 
aggregative,   and  whether  weighted  by  one  system  or 
another)  sinks  into  insignificance  as  compared  with  its 
conformity  to  the  two  tests.     The  only  things  which  are 
really  necessary  for  a  first  class  index  number  are : 

a.   Absence  of  freakishness ; 

6.   Conformity  to  Tests  1  and  2. 

The  conformity  to  Test  1  implies,  as  has  been  seen, 
absence  of  bias.  If  our  standards  of  a  good  index  number 
are  not  high,  we  need  not  insist  on  conformity  to  tests, 
but  instead  on  "  absence  of  bias." 

11.  Table  28  also  shows  that  Test  1  is  a  better  cor- 
rective of  bias  than  Test  2,  while  Test  2  is  a  better  cor- 
rective of  freakishness.     Thus,  as  a  rectification  of  the 
biased  arithmetic  Formula  7,  Formula  107  obeying  Test  1 
outranks  207  obeying  Test  2,  and  likewise  109  outranks 
209,    123   outranks   223,    125  outranks  225.     But  as  a 
rectification  of  the  freakish  median  33,  Formula  233  out- 
ranks 133,  and  235  outranks  135.     Again,  as  a  rectifi- 
cation of  the  freakish  simple  21,  Formula  221  outranks 
121 ;  while,  likewise,  231  outranks  131 ;  241  outranks  141 ; 
251  outranks  151. 

12.  The  most  accurate  formulae  are  those  toward  the 
end  of  the  list,  including  especially :  Formulas  353,  8053, 
2153,  1353,  1323,  5323. 

13.  If  the  data  for  quantities  are  available  only  for 
the  base  year  or  a  series  of  years,  the  best  available  index 
numbers  of  prices  are :  Formulae  53,  6053,  6023. 


COMPARING  ALL  THE  INDEX  NUMBERS    269 

14.  If   only   roughly   estimated    or   guessed   weights 
can  be  used,  the  best  formula  is  Formula  9051. 

15.  If  we  cannot,  or  will  not,  estimate  or  guess  at  the 
weights,  the  best  index  numbers  are :   Formulae  21,  101, 
31,  of  which  31  is  probably  slightly  more  accurate  unless 
there  is  good  reason  to  believe  that  the  true  weights  of 
the  various  commodities  really  are  approximately  equal, 
or  unless  the  number  of  commodities  is  very  small. 

We  may  restate  and  summarize  our  main  conclusions 
as  follows : 

Always  barring  the  mode  (as  a  freak  type)  and  the 
simple  (as  a  freak  weighting),  type  and  weighting  have  no 
material  influence  on*  our  final  results,  after  the  recfr'fi- 
cation  processes.  After  those  processes  are  completed, 
all  the  results  are  substantially  the  same.  This  will 
seem  a  startling  conclusion  and  quite  contrary  to  common 
opinion ;  for  current  views  do  not  recognize  the  existence 
of  bias  in  the  index  numbers  used  nor  realize  that  it  can 
be  rectified. 


CHAPTER  XIII 
THE  SO-CALLED  CIRCULAR  TEST 

§  1.  Introduction 

IT  will  be  remembered  that  the  fault  we  first  found  in 
certain  index  numbers,  e.g.  the  simple  arithmetic,  was 
that  it  would  not  work  consistently  as  between  two  times, 
or  between  two  places,  like  New  York  and  Philadelphia. 
Test  1  required  such  consistency  and  our  ideal  formula,  353, 
and  many  others  meet  that  test.  Can  we  and  ought  we 
to  extend  this  requirement  for  consistency  as  between 
the  two  times,  or  the  two  places,  which  the  index  num- 
ber compares  (and,  of  course,  it  only  compares  two)  to  a 
general  consistency  between  all  the  times  or  places  to 
which  we  apply  a  set  of  index  numbers  ? 

Hitherto  this  has  been  taken  for  granted  by  all  stu- 
dents of  index  numbers.  The  small  balls  ought,  it  has 
been  assumed,  always  to  lie  on  the  curve.  If  they,  or  any 
of  them,  are  separated  by  a  gap  from  the  curve,  then  it 
would  seem  there  must  be,  to  that  extent,  something 
wrong  in  the  index  number  which  permits  such  an 
inconsistency. 

By  the  so-called  "  circular  test,"  taking  New  York  as 
base  (=  100)  and  finding  Philadelphia  110,  then  taking 
Philadelphia  as  base  (=  110)  and  finding  Chicago  (115) 
we  ought,  when  we  complete  the  circuit  and  take  Chicago 
as  base  (=  115),  to  find,  by  direct  comparison,  New  York 
100  again.  Or  again,  if  Chicago  is  found  to  be  115  via 
Philadelphia,  it  ought  consistently  to  be  115  when  cal- 
culated directly. 

Still  again,  instead  of  taking  percentages,  let  us  take 

270 


THE  SO-CALLED   CIRCULAR  TEST  271 

easy  fractions.  Let  New  York  be  unity,  Philadelphia 
double  this  or  2,  Chicago  50  per  cent  more,  or  3.  Then 
New  York  should  be  (according  to  the  circular  test)  one 
third  of  Chicago,  or  1  again.  The  three  links  around 
the  circle  are  here  f  >  f ,  i,  and  these,  multiplied  together, 
give  unity  or  one  hundred  per  cent.1  For  a  single  com- 
modity, of  course,  this  holds  good.  If  the  price  of  sugar 
is  twice  as  high  in  Philadelphia  as  New  York,  50  per  cent 
higher  in  Chicago  than  Philadelphia,  then  self-evidently, 
in  New  York  the  price  of  sugar  must  be  a  third  as  high  as 
Chicago.  If  this  is  true  of  one  commodity,  why  not  of 
an  average  for  many? 

But  the  analogy  of  the  circular  test  with  the  time  re- 
versal test,  while  plausible,  is  misleading.  I  aim  to  show 
that  the  circular  test  is  theoretically  a  mistaken  one,  that 
a  necessary  irreducible  minimum  of  divergence  from 
such  fulfillment  is  entirely  right  and  proper,  and,  therefore, 
that  a  perfect  fulfillment  of  this  so-called  circular  test 
should  really  be  taken  as  proof  that  the  formula  which 
fulfills  it  is  erroneous. 

§  2.  Illustration  of  Non-fulfillment  by  Case  of 
Three  Very  Unlike  Countries 

We  can  see  best  by  a  concrete  example.  Let  us  take 
three  places  which,  to  fix  our  ideas,  we  shall  call  Georgia, 
Norway,  and  Egypt.  Take  a  list  of  15  commodities  of 
which  5,  led  by  lumber,  are  important  in  both  Georgia 
and  Norway;  5,  led  by  cotton,  are  important  in  both 
Georgia  and  Egypt ;  and  5,  led  by  paper,  are  important 
in  both  Egypt  and  Norway.  Let  us  further  suppose 
that  the  lumber  group,  important  in  both  Georgia  and 
Norway,  have  about  the  same  prices  in  Georgia  and  Nor- 

1  For  the  algebraic  expression  of  the  circular  test,  see  Appendix  I  (Note 
to  Chapter  XIII,  §1). 


272         THE  MAKING  OF  INDEX  NUMBERS 

way,  and  that  they  so  dominate  the  price  comparison 
between  these  two  countries  that  the  index  number  is 
about  the  same  in  both  countries,  the  other  two  groups 
of  commodities  in  these  two  countries  not  greatly  inter- 
fering with  this  equality,  because  one  is  unimportant  in 
Georgia  and  the  other  is  unimportant  in  Norway.  Like- 
wise, in  comparing  Georgia  and  Egypt,  the  cotton  group 
so  dominates  the  Georgia-Egypt  index  number  as  to 
make  Georgia  and  Egypt  about  the  same  price  level. 

We  might  conclude,  since  "  two  things  equal  to  the 
same  thing  are  equal  to  each  other, "  that,  therefore,  the 
price  levels  of  Egypt  and  Norway  must  be  equal,  and  this 
would  be  the  case  if  we  thus  compare  Egypt  and  Norway 
via  Georgia.  But  evidently,  if  we  are  intent  on  getting 
the  very  best  comparison  between  Norway  and  Egypt,  we 
shall  not  go  to  Georgia  for  our  weights.  In  the  direct 
comparison  between  Norway  and  Egypt  the  weighting  is, 
so  to  speak,  none  of  Georgia's  business.  It  is  the  concern 
only  of  Egypt  and  Norway.  In  such  a  direct  comparison 
between  Norway  and  Egypt,  the  paper  group,  which 
played  little  part  in  the  other  two  comparisons  now  tends 
to  dominate  the  situation ;  and  if  these  5  commodities  are 
higher  in  price  in  Norway  than  in  Egypt,  that  fact  may 
suffice  to  make  the  whole  Norwegian  price  level  some- 
what higher  than  the  Egyptian. 

§  3.   Comparisons  by  Index  Numbers  Differ 
in  Kind 

The  paradox  of  finding  the  price  levels  of  Norway  and 
Egypt  different,  although  by  separate  comparisons  the 
price  level  of  each  is  the  same  as  that  of  Georgia,  is  no 
more  strange  than  that  we  may  find  two  people  each 
resembling  in  their  features  a  third  person  without  re- 
sembling each  other.  Since  an  index  number  is  a  com- 


THE   SO-CALLED   CIRCULAR  TEST  273 

posite  dependent  on  heterogeneous  elements,  a  variation 
in  the  composition  will  change  the  comparison  qualita- 
tively. There  is  really,  therefore,  no  contradiction  or 
absurdity  in  the  apparent  inconsistencies ;  for  the  three 
comparisons  are  all  different  in  kind.  If  the  three  groups 
(lumber,  cotton,  paper)  prominent  in  the  Georgia, 
Norway,  Egypt  comparisons,  instead  of  merely  dominat- 
ing the  respective  comparisons,  were  completely  to  monop- 
olize them,  any  mystery  about  their  inconsistencies 
would  disappear.  We  would  have  three  index  numbers  of 
only  one  commodity  each :  lumber  for  comparing  Georgia 
and  Norway  (there  being  no  other  common  commodity), 
cotton  for  comparing  Georgia  and  Egypt  (this  being 
the  only  commodity  in  common),  and  paper,  the  only 
common  commodity,  for  comparing  Norway  and  Egypt. 
Our  supposedly  inconsistent  comparisons  reduce  to  the 
initial  facts,  viz.  that  lumber  is  the  same  price  in  Georgia 
as  in  Norway,  and  cotton  in  Egypt  as  in  Georgia,  while 
paper  is  higher  in  Norway  than  in  Egypt,  in  which  three 
statements  are  surely  no  mutual  inconsistencies.  The 
fact  that  lumber  and  cotton  show  certain  comparisons 
for  Norway  and  Egypt  relatively  to  a  third  country 
is  no  reason  why  a  commodity  quite  different  from  either 
lumber  or  cotton  should  show  any  particular  comparison 
between  Norway  and  Egypt  compared  directly.  Similarly, 
even  if  not  so  self -evidently,  the  fact  that  index  num- 
bers in  which  lumber  and  cotton  are  important  show 
certain  comparisons,  is  no  reason  why  an  entirely  dif- 
ferent index  number  in  which  they  are  unimportant  should 
show  any  particular  comparison. 

In  short,  each  dual  comparison  is  a  separate  problem 
differing  in  kind  from  every  other  and,  therefore,  requiring 
no  exact  correspondence  such  as  would  be  required  if  they 
were  not  different.  If  they  were  really  the  same,  e.g.  if  we 


274          THE  MAKING  OF  INDEX  NUMBERS 

had  one  and  the  same  commodity  to  deal  with,  it  would 
be  absurd  and  impossible  to  find,  say,  the  price  of  coffee  the 
same  in  Norway  as  in  Georgia,  the  same  in  Egypt  as  in 
Georgia,  but  yet  higher  in  Norway  than  in  Egypt. 

The  truth  is,  if  we  were  to  find  any  other  result  than 
what  we  have  found,  we  would  know  that  that  result  was 
wrong.  Such  a  formula  would  prove  too  much,  for  it  would 
leave  no  room  for  qualitative  differences.  Index  numbers 
are  to  some  extent  empirical,  and  the  supposed  inconsist- 
ency in  the  failure  of  (variably  weighted)  index  numbers  to 
conform  to  the  circular  test,  is  really  a  bridge  to  reality. 
That  is,  the  so-called  "  inconsistency  "  is  just  what  is 
needed  to  reconcile  our  theory  with  common  sense,  which 
tells  us  at  once  that  we  cannot  consistently  compare  far- 
distant  times  and  climes  by  means  of  averages  of  widely 
varying  elements.  Either  we  must  give  up  the  attempt, 
or  we  must  content  ourselves  with  an  artificially  rigid 
system  of  weights  which  contradicts  the  facts. 

§  4.  The  So-called  Circular  Test  can  be  Fulfilled 
Only  if  Weights  are  Constant 

The  only  formulae  which  conform  perfectly  to  the  cir- 
cular test  are  index  numbers  which  have  constant  weights, 
i.e.  weights  which  are  the  same  for  all  sides  of  the  "  tri- 
angle "  or  segments  of  the  "  circle,"  i.e.  for  every  pair 
of  times  or  places  compared.  Thus,  if  all  the  15  com- 
modities, lumber,  paper,  cotton,  etc.,  are  arbitrarily 
assigned  weights  which  remain  the  same  in  all  three 
comparisons,  in  defiance  of  the  actual  differences,  then 
the  index  number  ought  to  show  that  if  Norway  and 
Egypt  have  the  same  price  level  relatively  to  Georgia, 
they  will  have  the  same  price  level  relatively  to  each 
other.  And  this  is  precisely  what  we  do  find  of  the  simple 


THE  SO-CALLED  CIRCULAR  TEST  275 

or  constant  weighted  geometric,  for  instance,  and  the  sim- 
ple or  constant  weighted  aggregative.1 

But,  clearly,  constant  weighting  is  not  theoretically 
correct.  If  we  compare  1913  with  1914,  we  need  one  set 
of  weights;  if  we  compare  1913  with  1915  we  need, 
theoretically  at  least,  another  set  of  weights.  In  the 
former  case  we  need  weights  involving  the  quantities  of 
the  two  years  concerned,  1913  and  1914;  in  the  second 
case  we  need  weights  involving  the  (somewhat  different) 
quantities  of  the  two  years,  1913  and  1915.  We  cannot 
justify  using  the  same  weights  for  comparing  the  price 
level  of  1913,  not  only  with  1914  and  1915,  but  with  1860, 
1776,  1492,  and  the  times  of  Diocletian,  Rameses  II, 
and  the  Stone  Age ! 

Similarly,  turning  from  time  to  space,  an  index  number 
for  comparing  the  United  States  and  England  requires  one 
set  of  weights,  and  an  index  number  for  comparing  the 
United  States  and  France  requires,  theoretically  at  least, 
another.  To  take  extreme  cases,  it  would  obviously  be 
improper  to  use  the  same  weights  in  comparing  the  United 
States,  not  only  with  England  and  France,  but  with 
Russia,  Siberia,  China,  Thibet,  and  Central  Africa. 
In  comparing  hot  with  cold  climates,  coal  would  be 
weighted  heavily  in  some  cases  and  in  others  lightly, 
and  ice  reversely.  Allowances  should  likewise  be  made 
'  for  differences,  in  different  times  or  climes,  in  the  quanti- 
ties of  wool,  silk,  rice,  quinine,  ivory,  glass,  blubber, 
breadfruit,  sisal,  jade,  bamboo,  steel,  cement,  automobiles, 
boomerangs,  machine  guns,  linotype  machines,  wax 
tablets,  paper,  and  other  things  varying  in  importance 
geographically  or  historically.  In  comparing  the  prices 
of  our  times  with  those  of  1860,  it  is  just  as  important  to 
have  our  weights  representative  of  Lincoln's  day  as  to 

*  See  Appendix  I  (Note  A  to  Chapter  XIII,  §  4). 


276          THE  MAKING  OF  INDEX  NUMBERS 

have  them  representative  of  ours.  So  also  in  comparing 
our  country  with  China,  we  must  give  equal  voice  to  the 
peculiarities  of  the  two. 

If  we  start  with  weights  appropriate  to  the  United  States 
of  1922,  any  comparison  between  the  United  States  and 
modern  Kamchatka  or  ancient  Babylonia  would  be  one- 
sided. Even  more  one-sided  would  be  a  comparison, 
by  the  use  of  these  same  American  weights,  between  the 
price  levels  of  Kamchatka  and  Babylonia.  Only  by 
employing  the  weightings  of  the  United  States  in  1922, 
once  for  all,  are  we  enabled  to  force  a  fulfillment  of  the 
circular  test,  so  that  the  three  comparisons  between  the 
United  States  in  1922,  modern  Kamchatka,  and  ancient 
Babylonia  are  mutually  consistent.  For  instance,  if 
the  price  level  of  the  United  States  equalled  that  of  Kam- 
chatka and  also  equalled  that  of  Babylonia,  then  these 
two  would  equal  each  other.  It  is  clear  that  constant 
weighting,  though  it  makes  it  possible  to  fulfill  the 
circular  test,  does  so  at  the  expense  of  forcing  the  facts, 
for  the  true  weights  are  not  thus  constant.1 

§  5.  How  Closely  is  the  So-called  Circular 
Test  Fulfilled? 

But  the  important  question  is  :  How  near  is  the  circular 
test  to  fulfillment  in  actual  cases?  If  very  near,  then 
practically  we  may  make  some  use  of  the  circular  test  as 
an  approximation  even  if  it  is  not  strictly  valid.  To 
answer  this  question,  we  shall  take  Formula  353  and  the 
standard  set  of  data  for  1913-1918  which  we  have  used 
hitherto. 

Numerically,  by  Formula  353,  the  price  level  of  1914 

1  In  this  connection,  the  mathematical  reader  may  be  interested  in 
another  way  in  which,  with  a  limited  application,  the  circular  test  may  be 
fulfilled.  See  Appendix  I  (Note  B  to  Chapter  XIII,  §  4). 


THE  SO-CALLED  CIRCULAR  TEST 


277 


relatively  to  1913  is  100.12,  showing  a  rise  of  .12  per  cent. 
This  is  the  figure  obtained  by  comparing  the  two  years' 
prices  directly,  i.e.  without  the  intervention  of  any  other 
year.  But  if  we  compare  them  via  1915,  we  get  99.77 
for  1914,  showing  a  fall  of  .23  per  cent  from  1913  instead 
of  the  actual  rise  of .  12  per  cent.  The  following  table  gives 
all  the  comparisons  between  1913  and  1914,  both  directly 
and  also  indirectly,  via  certain  other  years. 


1913 

1914 

True  or  direct 

100 

100.12 

Indirect  via     1915 

100 

99.77 

1916 

100 

100.21 

1917 

100 

100.34 

1918 

100 

99.94 

It  will  be  noticed  that,  although  the  intervention  of  an 
intermediating  year  does  not  yield  exactly  the  same  result 
as  the  direct  comparison  between  the  two  years  concerned, 
the  discrepancies  are  very  slight.  This  is  found  to  be  true 
of  all  good  index  numbers.  That  is,  while  there  should  be 
some  discrepancy  and  the  index  numbers  which  have  none 
at  all  are  therefore  in  error,  a  large  discrepancy  is  equally 
wrong.  Formula  141,  for  instance,  exhibits  a  large 
discrepancy ;  353,  a  small  one. 

Let  us  test,  by  the  so-called  circular  test,  Formulae  9  and  353,  repre- 
senting a  very  bad  and  a  very  good  index  number  respectively ;  and,  for 
this  purpose,  let  us  take  the  circuit  of  years  1913-1914-1915-1913  or  0-1- 
2-0,  which  triangle  of  years  we  shall  refer  to  briefly  as  "012." 

By  Formula  9  the  index  number  for  the  side  of  the  triangle  0-1,  i.e. 
the  index  number  of  prices  for  1914  relatively  to  1913  as  base,  is  100.93 
per  cent ;  the  index  number  for  the  next  side  of  the  triangle,  1-2,  is  101 . 16 
per  cent;  and  that  for  the  returning  side,  2-0,  is  102.21  per  cent.  The 
product  of  these  three  index  numbers  around  the  triangular  circuit  is 
104.36  per  cent,  showing  that,  even  in  this  three-around  comparison,  the 
deviation  from  unity,  or  100  per  cent,  of  Formula  9  is  very  striking.  If 
we  should  take  a  four-around,  five-around,  or  six-around  case,  the  gap  in 


278         THE  MAKING  OF  INDEX  NUMBERS 

the  circle  would  be  much  greater.  Evidently  the  gap,  in  the  case  of  9,  is 
partly  due  to  its  known  upward  bias,  each  of  the  three  factors  tending  to 
be  larger  than  it  should  be. 

Next,  then,  let  us  try  Formula  353,  which  has  no  bias  and  fulfills  both 
tests.  In  this  case,  we  find,  for  the  same  circuit  0-1-2,  the  product  of  the 
three  index  numbers  for  prices,1  0-1,  1-2,  2-0,  is  100.35,  or  only  about  one 
third  of  one  per  cent  above  100  per  cent  or  unity.  The  other  index  num- 
bers, which  like  Formula  353  satisfy  both  Tests  1  and  2,  will,  in  general, 
deviate  from  the  so-called  circular  test  by  about  the  same  gap,  as  Table  33 
shows. 

TABLE  33.  THE  "CIRCULAR  GAP,"  OR  DEVIATION  FROM 
FULFILLING  THE  SO-CALLED  "CIRCULAR  TEST"  OF 
VARIOUS  FORMULA 

(In  the  3-around  comparison  of  price  indexes  for  years  1913-1914-1915, 

or  0-1-2) 


FORMULA  No. 

CIRCULAR  GAP 
(PER  CENTS) 

323 

+.34 

325 

+.38 

353 

+.35 

1323 

+.34 

1353 

+.34 

2353 

+.34 

5307 

+.40 

5323 

+.36 

This  table  shows  that  if  we  calculate  by  Formula  323,  starting  from 

1913  (year  0)  and  proceeding  to  1914  (year  1),  then  calculate  from  this 

1914  as  a  base  to  1915  (year  2),  and  then  calculate  from  this  1915  as  a  base 
to  1913  again,  instead  of  finding  ourselves  exactly  where  we  started,  the 
resulting  figure  will  be  slightly  above  the  starting-point,  exceeding  the 
original  figure  by  ^&  of  1  per  cent.     The  other  seven  formulae  give  almost 
uniformly  the  same  result,  roughly,  a  third  of  one  per  cent.     From  these 
examples,  and  others  which  will  be  noted  in  other  connections,  it  appears 
that  there  is  a  proper  and  there  is  an  improper  deviation  from  fulfillment 
of  the  circular  test.     The  deviation  or  circular  gap  of  about  one-third  of 
one  per  cent  for  Formula  353  and  other  good  formulae  represents,  as  it 
were,  an  irreducible  minimum  of  legitimate  deviation.     On  the  other 
hand,  the  big  gap  for  biased  formulae,  like  9,  represents,  for  the  most  part, 
an  illegitimate  or  erroneous  gap.     At  the  other  extreme,  the  simple  For- 
mulas 21  and  51  show  no  gap  at  all,  even  the  small  proper  deviation 
being  artificially  suppressed  by  the  use  of  constant  weighting. 

1  If  an  index  number  of  quantities  be  used,  the  circular  gap  will  be  equal 
but  opposite,  provided,  the  index  number  fulfills  Test  2.  See  Appendix  I 
(Note  to  Chapter  XIII,  §  5). 


THE  SO-CALLED  CIRCULAR  TEST 


279 


Circular  Test 

Gaps  for  Years  045 
of  formulae  1.9,23.141.151 


,41 


Vj  '/*  75  W  ' 

CHART  51.  Showing  that,  calculating  by  Formula  1  and  starting  from 
1913  (year  0),  then  proceeding  to  1917  (year  4),  1918  (year  5),  and  back  to 
1913,  the  year  from  which  we  started,  we  end  at  a  point  above  that  from 
which  we  started;  by  Formula  9  the  same  circuit  ends  still  higher;  by 
Formula  23  it  ends  lower;  by  Formula  141  (or  41)  it  ends  stilll  ower;  by 
Formula  151  (or  51)  it  ends  at  the  starting-point.  All  five  end  wrongly. 


280 


THE  MAKING  OF  INDEX  NUMBERS 


Graphically,  Chart  51  shows  five  formulae,  all  with  different  behaviours 
relatively  to  the  "circular  test,"  and  none  behaving  correctly.  Each 
relates  to  the  triangular  comparison  between  the  years  1913, 1917,  and  1918. 
Formula  1  is  far  from  conforming  to  the  circular  test,  returning  very  far 
above  the  starting-point.  Formula  9  returns  still  further  above,  23  returns 
to  1913  below  the  starting-point,  141  still  further  below,  while  151  returns 
exactly  to  the  starting-point. 

When  the  circular  test  is  fulfilled,  any  indirect  comparison  between,  say, 
1913  and  any  other  year,  say,  1915  via  1914,  will  agree  with  the  direct  com- 
parison ;  consequently,  the  chain  figures  will  coincide  with  the  fixed  base 
figures,  so  that  there  will  be  no  "  balls"  above  or  below  our  curves.  The 
more  nearly  the  circular  test  is  fulfilled,  the  more  nearly  will  the  balls  be 
to  the  curves.  Thus  the  reader,  by  studying  the  balls  in  relation  to  the 
curves  in  the  various  diagrams,  can  readily  gain  a  rough  idea  of  how  nearly 
the  circular  test  is  fulfilled.  This  subject  will  be  referred  to  again. 

§  6.   Complete  Tabulation  of  "  Circular  Gap  "  for 
Formula  363 

Table  34  gives  the  gaps  (for  353)  for  every  possible  triangle. 
TABLE   34.     THE    "CIRCULAR    GAP,"    OR    DEVIATION   FROM 


FULFILLING     THE    SO-CALLED    "CIRCULAR    TEST"    OF 
FORMULA   353    (IN  ALL  POSSIBLE  3-AROUND  COMPARI- 
SONS OF  PRICE  INDEXES) 

YEARS  OF 
"  TRIANGLE  " 

CIRCULAR  GAP 
(Pfcu  CENTS) 

0-1-2 

+.35 

0-1-3 

-.09 

0-1-4 

-.21 

0-1-5 

+.17 

0-2-3 

-.25 

0-2-4 

-.16 

0-2-5 

+.30 

0-3-4 

+.32 

0-3-5 

+.30 

0-4-5 

+.06 

1-2-3 

+.19 

1-2-4 

+.40 

1-2-5 

+.48 

1-3-4 

+.45 

1-3-5 

+.05 

1-4-5 

-.33 

2-3-4 

+.23 

2-3-5 

-.24 

2-4-5 

-.40 

3-4-5 

+.08 

THE  SO-CALLED   CIRCULAR  TEST 


281 


Even  the  maximum  of  these  circular  gaps  (that  for  the  triangle  of  the 
years  1-2-5,  or  1914-1915-1918-1914)  is  only  ^  per  cent,  or  less  than 
one  half  of  one  per  cent. 

We  find  the  same  smallness  of  the  gaps  when  the  "  circuit "  consists  of 
four  or  more  sides  around. 

Table  35  gives  all  the  quadrangular  or  4-around  comparisons. 


TABLE  35.  THE  "CIRCULAR  GAP,"  OR  DEVIATION  FROM 
FULFILLING  THE  SO-CALLED  "CIRCULAR  TEST"  OF 
FORMULA  353  (IN  ALL  POSSIBLE  4-AROUND  COMPARI- 
SONS OF  PRICE  INDEXES) 


YEARS  OF 
"  QUADRANGLE  " 

CIRCULAR  GAP 
(PER  CENTS) 

YEARS  OF 
"  QUADRANGLE  " 

CIRCULAR  GAP 
(PER  CENTS) 

0-1-2-3 

+.10 

0-3-2-4 

-.09 

0-1-2-4 

+.19 

0-3-2-5 

-.55 

0-1-2-5 

+.65 

0-3-^-5 

+.38 

0-1-3-2 

+.16 

0-3-5-4 

+.25 

0-1-3-4 

+.24 

0-4-1-5 

-.38 

0-1-3-5 

+.22 

0-4-2-5 

-.45 

0-1-4-2 

-.06 

0-4-3-5 

+.02 

0-1-4-3 

-.53 

1-2-3-4 

+.64 

0-1-4-5 

-.15 

1-2-3-5 

+.23 

0-1-5-2 

-.13 

1-2-4-3 

-.04 

0-1-5-3 

-.13 

1-2-4-5 

+.08 

0-1-5-4 

+.11 

1-2-5-3 

+.44 

0-2-1-3 

+.44 

1-2-5-4 

+.81 

0-2-1-4 

+.57 

1-3-2-4 

-.22 

0-2-1-5 

+.18 

1-3-2-5 

-.29 

0-2-3-4 

+.08 

1-3-4-5 

+.12 

0-2-3-5 

+.06 

1-3-5-4 

+.37 

0-2^-3 

-.48 

1-4-2-5 

-.07 

0-2-4-5 

-.10 

1-4-3-5 

+.41 

0-2-5-3 

-.00 

2-3-4-5 

-.17 

0-2-5-4 

+.24 

2-3-5-4 

+.15 

0-3-1-4 

+.13 

2-4-3-5 

+.48 

0-3-1-5 

-.26 

282 


THE  MAKING  OF  INDEX  NUMBERS 


Table  36  gives  all  the  5-around  comparisons. 

TABLE  36.  THE  "CIRCULAR  GAP,"  OR  DEVIATION  FROM 
FULFILLING  THE  SO-CALLED  "CIRCULAR  TEST"  OF 
FORMULA  353  (IN  ALL  POSSIBLE  5-AROUND  COMPARI- 
SONS OF  PRICE  INDEXES) 


YEARS  OF  S-AROUND 
CIRCUIT 

CIRCULAR  GAP 
(PER  CENTS) 

YEARS  OF  S-AROUND 
CIRCUIT 

CIRCULAR  GAP 
(PER  CENTS) 

0-1-2-3-4 

+.42 

0-2-4-3-5 

-.17 

0-1-2-3-5 

+  .41 

0-2-4-5-3 

-.40 

O-1-2-4-3 

-.13 

0-2-5-1-3 

-.04 

0-1-2-4-5 

+.25 

0-2-5-1-4 

+.09 

0-1-2-5-3 

+.35 

0-2-5-3-4 

+.32 

0-1-2-5-4 

+.59 

0-2-5-4-3 

-.08 

0-1-3-2-4 

+.01 

0-3-1-2-4 

+.28 

0-1-3-2-5 

+.46 

0-3-1-2-5 

+.74 

0-1-3-4-2 

+.39 

0-3-1-4-5 

+.07 

0-1-3-4-5 

+.29 

0-3-1-5-4 

-.20 

0-1-3-5-2 

-.08 

0-3-2-1-4 

+.32 

0-1-3-5-4 

+.16 

0-3-2-1-5 

-.07 

0-1-4^2-3 

-.30 

0-3-2-4-5 

-.15 

0-1^-2-5 

+.24 

0-3-2-5-4 

-.49 

0-1-4-3-2 

-.28 

0-3-4-1-5 

-.70 

0-1^1-3-5 

-.23 

0-3-4-2-5 

-.77 

0-1-4-5-2 

-.45 

0-3-5-1-4 

+.08 

0-1^-5-3 

-.46 

0-3-5-2-4 

+.15 

0-1-5-2-3 

-.38 

0-4-1-2-5 

+.87 

0-1-5-2-4 

-.28 

0-4-1-3-5 

+.43 

0-1-5-3-2 

+.12 

0-4-2-1-5 

+.02 

0-1-5-3-4 

+.19 

0-4-2-3-5 

+.21 

0-1-5-4-2 

+.27 

0-4-3-1-5 

+.07 

0-1-5-4-3 

-.21 

0-4-3-2-5 

-.22 

0-2-1-3-4 

+.12 

-2-3-4-5 

+.31 

0-2-1-3-5 

+.14 

-2-3-5-4 

+.56 

0-2-1-4-3 

+.89 

-2-4-3-5 

+.01 

0-2-1-4-5 

+.51 

-2-4-5-3 

+.04 

0-2-1-5-3 

+.48 

-2-5-3-4 

+.89 

0-2-1-5-4 

+.24 

-2-5-4-3 

+.36 

0-2-3-1-4 

+.38 

1-3-2-4-5 

+.11 

0-2-3-1-5 

-.01 

1-3-2-5-4 

-.61 

0-2-3-4-5 

+.13 

1-3-4-2-5 

-.52 

0-2-3-5-4 

-.00 

1-3-5-2-4 

+.03 

0-2-4-1-3 

+.03 

1-4-2-3-5 

-.17 

0-2-4-1-5 

-.23 

1-4-3-2-5 

+.16 

THE  SO-CALLED  CIRCULAR  TEST 


283 


Table  37  gives  all  the  6-around  comparisons. 

TABLE  37.  THE  "CIRCULAR  GAP,"  OR  DEVIATION  FROM 
FULFILLING  THE  SO-CALLED  "CIRCULAR  TEST"  OF 
FORMULA  353  (IN  ALL  POSSIBLE  6-AROUND  COMPAR- 
ISONS OF  PRICE  INDEXES) 


YEARS  OF  G-AROUND 
CIRCUIT 

CIRCULAR  GAP 
(PER  CENTS) 

YEARS  OF  G-AROUND 
CIRCUIT 

CIRCULAR  GAP 
(PER  CENTS) 

0-1-2-3-4-5 

+.48 

0-2-3-1-4-5 

+.32 

0-1-2-3-5-4 

+.35 

0-2-3-1-5-4 

+.05 

0-1-2-4-3-5 

+.18 

0-2-3-4-1-5 

-.45 

0-1-2^-5-3 

-.05 

0-2-3-5-1-4 

+.33 

0-1-2-5-3-4 

+.67 

0-2-4-1-3-5 

-.27 

0-1-2-5-4-3 

+.27 

0-2-4-1-5-3 

+.08 

0-1-3-2-4-5 

+.06 

0-2-4-3-1-5 

+  .22 

0-1-3-2-5-4 

+.41 

0-2-4-5-1-3 

+.36 

0-1-3-4-2-5 

+.69 

0-2-5-1-3-4      i 

-.36 

0-1-3-4-5-2 

-.01 

0-2-5-1-4-3 

+.41 

0-1-3-5-2-4 

-.24 

0-2-5-3-1-4 

+.13 

0-1-3-5-^2 

+.32 

0-2-5-4-1-3 

-.36 

0-1-4-2-3-5 

-.00 

0-3-1-2-4-5 

+.34 

0-1-4-2-5-3 

-.06 

0-3-1-2-5-4 

+.69 

0-1-4-3-2-5 

+.02 

0-3-1-4-2-5 

+.34 

0-1-4-3-5-2 

-.53 

0-3-1-5-2-4 

-.19 

0-1-4-5-2-3 

-.70 

0-3-2-1-4-5 

+.26 

0-1-4-5-3-2 

-.21 

0-3-2-1-5-4 

-.01 

0-1-5-2-3-4 

-.05 

0-3-2-4-1-5 

-.47 

0-1-5-2-4-3 

-.60 

0-3-2-5-1-4 

-.16 

0-1-5-3-2-4 

-.04 

0-3-4-1-2-5 

+  1.19 

0-1-5-3-4-2 

+.C5 

0-3-4-2-1-5 

-.30 

0-1-5-4-2-3 

+.02 

0-3-5-1-2-4 

+.33 

0-1-5-4-3-2 

+.04 

0-3-5-2-1-4 

+.56 

0-2-1-3-4-5 

+.06 

0-4-1-2-3-5 

+.62 

0-2-1-3-5-4 

+.20 

0-4-1-3-2-5 

+.68 

0-2-1-4-3-5 

+.59 

0-4-2-1-3-5 

-.02 

0-2-1^-5-3 

+.81 

0-4-2-3-1-5 

-.16 

0-2-1-5-3-4 

+.16 

0-4-3-1-2-5 

+.42 

0-2-1-5-4-3 

+.56 

0-4-3-2-1-5 

+.25 

§  7.   Discussion  of  the  "Circular  Gap"  of 
Formula  353 


Tables  34-37  give  all  the  possible  circuits  among  the 
years  1913-1918,  and  the  "  gap  "  found  for  each  circuit 
according  to  Formula  353.  As  we  have  seen,  these  devia- 


284         THE  MAKING  OF  INDEX  NUMBERS 

tions  are  normal  phenomena,  not  errors,  but  fortunately 
they  are  so  small  that  for  practical  purposes  they  are  not 
worth  taking  into  account.  The  maximum  gap  among  all 
the  20  possible  triangular  comparisons  is,  as  already  noted, 
only  .48  per  cent  (for  the  circuit  of  the  three  years  1-2-5). 
The  maximum  gap  among  all  the  45  possible  quadrangular 
circuits  is  .81  per  cent  (for  the  years  1-2-5-4).  The  maxi- 
mum for  the  72  5-around  comparisons  is  .89  per  cent  (for 
0-2-1-4-3  or  1-2-5-3-4).  Lastly,  the  maximum  for  the 
60  6-arounds  is  1.19  per  cent  (for  the  years  0-3-4-1-2-5). 

Even  these  gaps  are  unusually  large.  By  the  ex- 
pression for  the  "  probable  deviation "  we  estimate 
that  if  any  one  of  the  20  3-around  figures  be  selected  by 
lot,  it  is  as  likely  as  not  that  it  will  be  less  than  .19  per 
cent ;  while,  a  like  random  choice  among  the  45  4-arounds 
will,  as  likely  as  not,  be  less  than  .22  per  cent ;  of  the  5- 
arounds,  .25  per  cent ;  and  of  the  6-arounds,  .27  per 
cent.  In  a  word,  the  circular  test  is  generally  fulfilled 
within  one  fourth  of  one  per  cent ! 

The  maximum  gap  and  the  probable1  gap  for  each 
group  are  given  in  Table  38. 

TABLE  38.     "CIRCULAR  GAPS"   FOR  FORMULA  353 


MAXIMUM  (PER  CENTS) 

PROBABLE  (PER  CENTS) 

3-around 

.48 

.19 

4-around 

.81 

.22 

5-around 

.89 

.25 

6-around 

1.19 

.27 

Even  these  infinitesimal  results  need  to  be  divided  in 
several  pieces  to  give  the  share  of  the  deviation  per- 

1  That  is,  the  gap  which  is  exactly  as  likely  as  not.  This  is  the  usual 
sense  employed  in  studies  of  probability,  i.e.  the  "probable  error"  of 
the  series,  i.e.  .6745  X  the  standard  deviation,  or  square  root  of  the  average 
square. 


THE  SO-CALLED   CIRCULAR  TEST  285 

taining  to  any  individual  index  number,  for  it  is  to  be 
remembered  that  the  3-around  gap  is  to  be  distributed 
among  the  three  sides  of  the  triangle  so  that  to  suppress 
a  .19  per  cent  gap  entirely  and  force  a  complete  fulfillment 
of  the  circular  test,  it  would  be  necessary  to  "  doctor  " 
each  of  the  three  index  numbers  by  only  .06  of  one  per 
cent! 

Furthermore,  the  case  we  are  considering  of  36  commodi- 
ties, very  widely  dispersing  in  war-disturbed  years,  is  a 
very  extreme  and  unusual  case.  In  ordinary  times  the 
gap  would  be  even  less,  and  this  would  be  true  even  if  a 
great  number  of  years  were  taken.  Each  additional 
year  in  the  circuit  at  first  increases  the  probable  gap,  in 
the  extreme  case  here  considered,  by  about  .03 ;  at  this 
rate  without  allowing  for  any  diminution,  it  would  require 
a  full  century  probably  to  bring  the  circular  test  gap  up 
to  three  per  cent !  And  this  is  a  conservative  figure ;  for 
the  gap  increases  with  the  dispersion  and,  as  has  been 
often  noted,  the  dispersion  of  our  36  commodities  during 
this  war  period,  1913-1918,  is  much  greater  than  usual. 

Sauerbeck's  data  (for  36  commodities  selected  as  nearly 
like  our  36  as  possible)  show  a  dispersion  between  1846 
and  1913,  a  period  of  67  years,  of  only  42.10  per  cent,  or 
less  than  that  (45.09  per  cent)  of  our  36  commodities  in  five 
years.  It  follows,  therefore,  that,  had  Sauerbeck  been 
able  to  use  Formula  353,  the  discrepancy  between  the 
fixed  base  and  chain  system  would  have  been  found  to  be 
in  67  years  less  than  the  .27  per  cent  for  our  36  commodi- 
ties in  five  years,  say,  less  than  }  of  one  per  cent  and  less 
than  |  of  one  per  cent  for  a  century  consisting  of  years 
no  more  disturbed  than  the  67  mentioned ;  but  apparently 
the  addition  to  the  gap  gradually  diminishes,  so  that  it 
would  really  be  even  less.  It  follows  that,  except  for  very 
long  periods  or  for  periods  of  greater  dispersion  than  the 


286         THE  MAKING  OF  INDEX  NUMBERS 


Circular   Test 

Largest    Gaps   For 

5-Around 
4 -Around 
5 -Around 
6- Around 


IP* 


73  74  75  7<?  77  78 

CHART  52.  The  circular  test  gap  (at  the  left  of  each  of  the  four  circuits), 
even  at  its  greatest,  as  here  charted  for  Formula  353,  is  remarkably  small 
in  all  cases.  It  slightly  increases  as  the  circuit  of  year-to-year  index  num- 
bers becomes  more  circuitous,  reaching  over  one  per  cent  in  the  6-around 
circuit,  1913-'16-'17-'14-'15-'18-13. 

years  of  the  World  War,  if  such  be  possible,  or  both,  the 
circular  test  is  always  satisfied  by  the  ideal  Formula  353 
for  all  intents  and  purposes. 


THE  SO-CALLED   CIRCULAR  TEST  287 

Graphically,  the  four  maximum  gaps  for  Formula  353 
are  given  in  Chart  52.  The  lines  return  so  nearly  to  the 
starting  point  in  each  case  that  the  observer  has  to  look 
closely  to  see  the  gap.  The  "  probable  "  gap  is  not 
pictured  but  would  be  in  all  cases  about  half  the  .48  per 
cent  gap  in  the  chart,  the  maximum  for  the  3-around 
comparison. 

§  8.   Comparing  the  Circular  Gaps  of  the  134 
Different  Formulae 

Since  the  circular  gap  is  the  proper  and  necessary  result 
of  the  ceaseless  changing  of  the  weights  in  our  year-to- 
year  comparisons,  it  is  interesting  to  note  that,  among  the 
best  types  of  index  numbers,  the  various  gaps  roughly 
correspond. 

Since  no  other  index  number  has  been  worked  out  for  all  possible  com- 
parisons as  was  Formula  353,  we  cannot  study  other  formulae  by  exactly 
the  same  methods  as  we  have  just  studied  353.  The  only  comparisons 
available  are  those  furnished  by  the  contrasts  between  the  ordinary  fixed 
base  and  chain  index  numbers. 

Graphically,  in  Chart  49,  the  little  vertical  black  lines  (as  explained  in 
detail  in  the  fine  print  below  the  chart)  measure  the  deviations  of  each  point 
from  the  position  it  would  occupy  had  it  fulfilled  the  circular  test.  Near  the 
end  of  the  list  in  Chart  49,  the  balls  have  substantially  the  same  relative 
positions  for  all  the  curves,  as  do  also  the  tiny  vertical  dark  lines  indicating 
the  year-to-year  deviations  under  the  circular  test.  We  have  to  count 
off  nearly  40  curves  (from  the  "ideal"  at  the  bottom)  before  we  reach 
one  which  shows  an  appreciable  difference  in  the  position  of  its  balls. 
Beyond  this  point,  as  we  encounter  the  less  exact  formulas,  we  find  an 
increasing  variability  of  the  position  of  the  last  ball  which  never  again 
sits  close  on  the  curve  as  in  Formula  353  and  the  neighboring  curves. 

There  are  three  ways  or  methods  by  which  the  eye  can  sense  the  degree 
of  deviation  of  the  four  balls  from  any  curve.  The  first  and  easiest  is 
merely  to  note  the  position  of  the  last  ball,  i.e.  that  for  1918,  which  expresses 
the  net  cumulative  result  of  all  four  deviations.  But  this  method  gives 
merely  the  final  result  and  ignores  the  intervening  history.  The  four 
successive  deviations,  like  four  successive  tosses  of  a  coin,  will  occasionally 
(once  in  16  rounds),  all  accumulate  in  one  direction;  on  the  other  hand, 
though  all  four  deviations  may  be  great,  they  may  happen  largely  to 
offset  each  other. 

The  second  way  of  reading  the  deviations,  therefore,  is  to  run  the  eye 
over  all  four  balls  and  note,  in  a  general  way,  how  far  they  vary  from  the 


288         THE  MAKING  OF  INDEX  NUMBERS 

curve.  For  curves  near  the  bottom  of  Chart  49,  the  two  methods  show 
the  same  results,  but  for  curves  near  the  top  they  show  some  different 
results.  The  second  method  also  may  sometimes  give  an  incomplete 
picture.  For  instance,  as  between  the  two  curves  —  that  for  the  fixed 
base  drawn  in  black  and  that  which  we  imagine  as  connecting  the  balls  — 
the  only  disagreement  may  be  all  in  the  second  link,  1914-1915.  After  that 
point  the  curves  may  run  exactly  parallel ;  in  which  case,  the  second,  third, 
and  fourth  balls  inherit  the  exact  deviation  of  the  first  and  the  eye  will 
be  apt  to  count  this  one  deviation  four  times,  —  in  Charts  48,  eight  times. 
It  is  clear  that  the  proper  way  to  measure  the  four  deviations  is  the 
third  way,  namely,  to  examine  each  separately  as  a  year-to-year  matter. 
This  is  indicated,  in  Chart  49,  by  the  vertical  dark  broad  line.  This 
line  shows,  not  how  far  the  ball  is  from  the  curve,  but  how  much  farther 
or  nearer  it  is  than  the  preceding  ball.  If  a  ball  is  in  exactly  the  same  position 
relatively  to  the  curve  as  the  preceding  ball,  —  if,  for  instance,  they  are  both 
just  a  quarter  of  an  inch  below  the  curve,  —  there  will  be  no  dark  line.  It 
is  the  displacement  from  this  position  which  the  dark  line  measures ;  that 
is,  the  extent  to  which  the  chain  figure  has  gotten  out  of  line  in  either  direc- 
tion since  the  last  year.1 

The  eye  can  readily  sense  the  totality  of  these  black 
lines  for  any  curve  and  compare  that  totality  with  that  for 
any  other  curve.  It  requires  only  a  glance  at  Chart  49 
to  see  that  the  "  worthless  "  and  "  poor  "  index  numbers 
have  the  dark  lines  very  much  in  evidence  except  in  a 
few  cases  (where  they  are  made  to  disappear  entirely 
by  artificially  assuming  the  weights  constant).  The 
11  fair  "  index  numbers  show  less  blackness ;  the  "  good  " 
still  less ;  the  "  very  good  "  very  much  less.  The  "  excel- 
lent "  still  less  and  the  "  superlative"  the  least  of  all  — 
so  little,  in  fact,  as  scarcely  to  be  perceptible  to  the  eye. 
And  this  seems  reasonable.  For  while,  as  we  have  seen, 
there  must  be  some  deviation  to  express  truly  the  effect  of 
varied  weighting,  we  have  found  the  effect  really  negligible. 

§  9.  Status  of  all  Formulae  Relatively  to  the 
So-called  Circular  Test 

So  negligible  is  this  normal  gap  as  compared  with  the 
ordinary  effects  of  bias  or  freakishness,  that  when  these 

i  See  Appendix  I  (Note  to  Chapter  XIII,  §  8). 


THE  SO-CALLED  CIRCULAR  TEST 


289 


effects  are  present  they  dominate.  Thus  we  have  three 
chief  cases  to  distinguish :  (1)  where  bias  or  freakishness 
is  responsible  for  the  gap ;  (2)  where  the  gap  is  forcibly 
suppressed  by  constant  weighting,  and  (3)  the  remaining 
cases  where  the  gap  is  normal. 

TABLE  39.    LIST    OF    FORMULA    IN    (INVERSE)    ORDER    OF 
CONFORMITY    TO    SO-CALLED    CIRCULAR    TEST 


FORMULA 
No. 

RANK 

FORMULA 
No. 

RANK 

FORMULA 
No. 

RANK 

43 

35 

27 

15 

125 

5 

201 

34 

37 

" 

126 

M 

243 

33 

237 

44 

227 

II 

245 

1 

14 

325 

M 

247 

209 

44 

1104 

M 

249 

333 

41 

1153 

4t 

44 

i2 

2 

13 

1154 

4i 

46 

10 

14 

1303 

M 

48 

207 

44 

2154 

" 

50 

1333 

44 

3154 

II 

41=141 

31 

11 

12 

3353 

M 

9 

30 

211 

44 

4153 

•• 

35 

29 

233 

« 

5307 

•• 

1133 

" 

335 

44 

54** 

4 

13 

28 

14 

11 

301 

M 

15 

27 

30 

44 

309 

M 

7 

26 

235 

44 

353  f 

«• 

12 

25 

5333 

44 

1014 

<« 

32  =  132 

24 

16 

10 

1124 

•« 

40 

23 

225 

14 

1353 

it 

38 

22 

223 

9 

2353 

it 

39 

44 

229 

44 

5323 

i* 

31=131 

21 

231  =331 

ii 

8053 

«« 

241  =341 

" 

8 

8 

8054 

«« 

33 

20 

24 

44 

101 

3 

34 

44 

53* 

44 

1123 

•i 

135 

M 

102 

44 

1323 

ii 

215 

44 

108 

44 

2153 

ii 

1013 

41 

1004 

44 

123 

2 

239 

19 

26 

7 

323 

•• 

25 

18 

28 

44 

21=121 

1 

133 

«• 

307 

44 

22  =  122 

ii 

134 

44 

109 

6 

51=151 

ii 

136 

44 

124 

44 

52  =  152 

ii 

213 

44 

1103 

44 

221  =321 

ii 

23 

17 

3153 

44 

251=351 

i* 

1134 

44 

4154 

44 

6023 

•• 

29 

16 

4353 

44 

6053 

«« 

36 

44 

107 

5 

9021 

44 

42=142 

44 

110 

14 

9051 

ii 

1003 

*53  =  3  =  6=  17  =  20  =  60. 
**  54  =4=5  =  18=19=59. 
1 353  =  103  - 104  - 105  =  106  - 153  -154  -203  =205  =217  =219  =253  =259  =303  =305. 


290 


THE  MAKING  OF  INDEX  NUMBERS 


The  formulae  in  class  2,  —  those  conforming  to  the  test  by  force,  so  to 
speak,  are  121  (  =  21),  122  (  =  22),  151  (=51),  152  (=52),  321  (=221), 
351  ( =251),  6023,  6053,  9021,  9051,  only  ten  l  formulae  in  all,  all  geometries 
and  aggregatives.  Those  in  class  3  can  be  set  off  less  definitely  as  the 
gradations  are  so  gradual.  Practically,  however,  they  are  identical  with 
the  "superlative"  group  which  we  set  apart  —  also  somewhat  arbitrarily 
—  on  the  score  of  nearness  to  the  ideal,  Formula  353. 

In  Table  39  the  formulae  are  roughly  ranked  solely  according  to  the 
degree  of  conformity  to  the  so-called  circular  test.2 

From  this  table  it  is  clear  that  (excepting  those  at  the  bottom  of  the 
list  which  hold  their  rank  unfairly,  by  stereotyped  weights)  Formula  353 


Dispersion 

(Measured  by  Standard  Deviations) 
(Prices,  Fixed  Base) 


353 


75  '/*  75  ^  *7  VB 

CHART  53  P.    Showing  the  average  dispersion  of  the  36  price  relatives 
taken  relatively  to  the  fixed  base,  1913,  on  either  side  of  the  ideal  (353). 

1  Not  counting  Formula  7053  (discussed  in  the  next  chapter)  which 
might  be  added  to  the  list,  although  on  a  slightly  different  basis. 

2  The  rank  of  each  is  reckoned  roughly  by  adding  together  the  dark 
lines  in  Chart  49  (after  first  applying  to  the  several  lines  for  the  several 
years  rough  equalizing  coefficients  based  on  the  standard  deviations  of 
the  36  commodities  somewhat  on  the  analogy  of  the  method  used  for 
reckoning  the  order  of  merit  or  accuracy  in  Table  28). 


THE  SO-CALLED  CIRCULAR  TEST 


291 


and  its  former  rivals  hold  close  to  first  place  here  also ;  and  that,  with  few 
exceptions,  the  ranking  here  corresponds  roughly  to  the  former  ranking  in 
respect  of  nearness  to  353.  This  confirms  Walsh's  conclusion  on  the  same 
subject  on  the  basis  of  which  he  accorded  the  first  prize  to  353. l 

Thus,  we  find  that  theoretically  and  practically  the  best  formulae  should 
not  and  do  not  yield  index  numbers  which  will  check  perfectly  when  the 
circular  test  is  applied.  It  is  true  that  the  best  forms  of  index  numbers, 
as  determined  by  other  standards,  usually  check  more  closely  under  this 
test  than  do  the  poorest.  This  is  not,  however,  because  the  circular  test 
is  a  valid  test  of  good  index  numbers  for  it  is  not,  but  merely  because  any 
large  defects  of  a  formula  which  would  classify  it  as  a  poor  one  under 
Tests  1  and  2  are  likely  to  classify  it  as  a  poor  one  under  the  circular  test. 

In  fact,  the  effects  of  the  change  in  the  relative  weights  of  different 


Dispersion 

(Measured  by  Standard  Deviations) 
(Quantities.  Fixed  Base) 


353 


'& 


75  76  '/7 

CHART  53Q.    Analogous  to  Chart  53P. 


73 


commodities  make  themselves  felt  so  slowly  that  the  best  formulae  yield 
results  which  check  under  the  circular  test  to  a  degree  of  accuracy  far  be- 
yond that  required  for  any  practical  use  to  which  index  numbers  are  now 
put.  In  other  words,  this  means  that  a  single  series  of  index  numbers 
(i.e.  one  index  number  for  each  year)  which  is  calculated  by  any  one  of 
the  best  formulae  will  permit  the  comparison  of  price  levels  of  any  two 
years  to  a  degree  of  accuracy  beyond  anything  which  is  likely  to  be  re- 
quired for  practical  purposes. 

Practically,  then,  the  test  maybe  said  to  be  a  real  test. 
Theoretically  it  is  not  ;  for  the  ranking  of  formulae  ought, 

1  The  Problem  of  Estimation,  p.  102. 


292 


THE  MAKING  OF  INDEX  NUMBERS 


in  strictness,  to  be  relative  not  to  a  perfect  fulfillment  of 
the  test  but  to  the  irreducible  minimum  exhibited  by 
Formula  353  (or  its  peers).  That  is,  we  should  condemn 
the  ten  formulae  which  close  the  gaps  entirely  just  as  truly 
as  those  where  the  gap  is  larger.  Thus  the  test  is  not 
an  essential  one  in  the  theory  of  index  numbers.1 


Dispersion 

(Measured  by  Standard  Deviations) 
(Prices.  Chain) 


73  7+  75  ?G  17  78 

CHAET  54P.     Showing  the  average  dispersion  of  the  price  relatives  taken 
each  year  relatively  to  the  preceding  year,  chain  fashion. 

§  10.  Macaulay's  and  Ogburn's  Theorem 

Professor  Frederick  R.  Macaulay,  referring  to  arithmetic  index  numbers, 
says : 2  "the  chain  numbers  draw  away  (upwards)  from  the  fixed  base  num- 
bers" because  of  a  "greater  tendency  to  rise  and  a  less  tendency  to  fall 
(in  percentages)  with  the  smaller  relatives  than  with  the  larger  relatives." 

1  There  are  other  and  still  less  essential  tests  which  might  be  considered 
and  were  discussed  by  me  in  my  Purchasing  Power  of  Money  (Appendix 
to  Chapter  X).  See  Appendix  I  (Note  to  Chapter  XIII,  §  9). 

8  American  Economic  Review,  March,  1916,  p.  208. 


THE  SO-CALLED  CIRCULAR  TEST 


293 


Macaulay  verifies  this  conclusion  by  actual  instances.  It  is  also  confirmed 
by  the  present  study,  for  we  find  that  the  typically  arithmetic  index  numbers, 
Formula  1  (the  simple)  and  Formula  1003  (the  cross  weighted)  as  well 
as  7  and  9  show  a  cumulative  upward  tendency  of  the  balls.1  Ma- 
caulay's  and  Ogburn's  same  reasoning  could  be  applied  reversely  to  the 
harmonic  to  show  accumulation  downward.  This  is  illustrated  by  For- 
mula 11,  13,  15,  1013. 

The  principle  involved  may  be  stated  in  this  form :  the  chain  arithmetic 
has  a  greater  upward  bias  than  the  fixed  base  arithmetic ;  while,  likewise, 
the  chain  harmonic  has  a  greater  downward  bias  than  the  fixed  base  har- 
monic. 


Dispersion 

(Measured  by  Standard  Deviations) 
(Quantities.  Chain) 


353 


73  '14  75  7<?  17  78 

CHART  54Q.    Analogous  to  Chart  54P. 

Graphically,  there  is  a  simple  way  of  picturing  this  principle.  We  have 
seen  that  where  there  is  bias  in  a  price  index,  this  bias  increases  rapidly  with 
the  dispersion  of  the  price  relatives.  The  reason  the  bias  of  the  chain 
system  increases  faster  than  that  of  the  fixed  base  system  is  that  the  dis- 
persion in  the  chain  system  increases  faster  than  in  the  fixed  base  system. 
This  fact  is  evident  from  Charts  53  P  and  53Q  which  show  that  the  stand- 
ard deviation  on  the  fixed  base  system,  while  it  increases  with  the  years, 
increases  more  and  more  slowly.  The  dispersion  starts  off  with  a  spurt, 
the  first  two  lines  diverging  from  the  curve  at  a  big  angle.  But  year  by 
year  (in  general)  the  angle  (relatively  to  the  central  curve)  diminishes.  With 
the  chain  system,  however,  a  new  start  is  made  every  year  so  that  we 
have  a  succession  of  spurts  with  no  subsequent  tendency  to  slow  up  as  in 
the  fixed  base  system.  Each  line  in  Charts  54P  and  54Q  for  the  standard 


1  Professor  William  F.  Ogburn  has  shown  this  algebraically,  on  the  basis 
of  probability  theory.    See  Appendix  I  (Note  to  Chapter  XIII,  §  10). 


294         THE  MAKING  OF  INDEX  NUMBERS 

deviation  has  a  slope  diverging  from  the  curve  at  an  angle  greater  than  the 
corresponding  line  for  that  same  year  in  the  fixed  base  system  of  Chart  53. 
The  same  slowing  up  is  seen  in  Chart  55  which  shows  the  dispersion  for 
Sauerbeck's  index  number  of  prices,  the  dispersion  being  reckoned  rela- 
tively to  the  earliest  year,  1846,  as  fixed  base.1 

Dispersion 
(Measured  by  Standard  Deviations) 

(Prices,  fixed  Base) 
(Sauerbe&S  Figures) 


'46        '56  V6          76  '86  96  V6       13 

CHART  55.  Showing  the  average  dispersion  of  36  of  Sauerbeck's  price 
relatives,  analogous  to  the  36  of  this  book,  taken  relatively  to  a  fixed  base, 
1846.  The  dispersion  in  the  five  years,  1913-1918  (shown  in  Chart  53P) 
exceeds  the  dispersion  shown  in  this  Chart  for  67  years. 

In  short,  the  acceleration  of  the  chain  bias  is  due  to  the  retardation  of 
the  fixed  base  .dispersion.  The  same  tendency  for  the  dispersion  on  the 
fixed  base  system  to  slow  up  as  time  goes  on  may,  of  course,  be  shown  by 
the  method  of  "quartiles"  or  "deciles"  relatively  to  the  median.  The 

Sauerbeck's  index  number  itself  is  on  the  base  1867-1877.  These 
charts  may  also  be  used  in  connection  with  the  discussions  on  bias,  in 
relation  to  dispersion,  of  Chapter  V. 


THE  SO-CALLED  CIRCULAR  TEST  295 

many  curves  of  this  sort  worked  out  by  Wesley  C.  Mitchell  show  this  slow- 
ing up  tendency  clearly.1 

§  11.  The  "  Circular  Test "  Reduced  to  a 
"  Triangular  Test " 

Before  leaving  the  so-called  circular  test,  it  may  be 
worth  while  to  note  that  it  may  be  considered,  at  bottom, 
to  be  simply  a  triangular  test.  If  any  formula  (besides 
satisfying  the  time  reversal  test)  will  satisfy  the  circular 
test  for  any  3-around  circuit  it  will  necessarily  satisfy  it 
for  a  4-around,  5-around,  or  any  other  larger  number  of 
steps.  This  extension  beyond  the  original  three  is  easily 
proved.2 

§  12.  Historical 

The  basic  idea  of  the  circular  test  was  first  explicitly 
propounded  by  Westergaard,  who  maintained  that  a 
change  in  the  base  ought  not  to  affect  the  relative  sizes 
of  the  index  numbers  of  the  different  years.  Walsh,  in 
his  Measurement  of  General  Exchange  Value,  greatly 
emphasized  this  idea.  He  expresses  it  in  the  slightly 
modified  form  which,  afterward,  in  his  Problem  of 
Estimation,  he  called  the  "  circular  test."  He  took  the 
ground  that,  like  other  tests  taken  individually,  it  is  of 
itself  only  negative,  capable  of  disproving,  but  not  of 
proving,  an  index  number.  He  noted  that  several  old 
and  familiar  formulae,  obviously  faulty  for  their  failure 
to  fulfill  other  and  simpler  tests,  completely  conform  to 
this  one.  The  only  formulae  which  he  found  to  conform 
perfectly  had  constant  weights.3  He  sought  for  such  con- 

1  Wesley   C.  Mitchell,  Business   Cycles,   pp.    Ill,    137,   University  of 
California  Press,  1913. 

2  See  Appendix  I  (Note  to  Chapter  XIII,  §  11). 

3  See  Walsh,  Measurement  of  General  Exchange  Value,  pp.  334,  335,  393, 
397,  398,  399,  431. 


296         THE  MAKING  OF  INDEX  NUMBERS 

formity  among  the  formulae  recommendable  for  reasons 
derived  from  the  study  of  the  nature  of  exchange  values 
and  of  averages,  but  he  was  unable  to  find  any  formulae 
that  accurately  satisfy  this  test. 

Among  the  formulae  which,  for  such  reasons,  he  could 
recommend,  he  counted  as  best  those  which  came  nearest 
to  satisfying  this  test.  His  latest  conclusion  is  that  the 
formula  which  I  have  called  "  ideal  "  comes  nearest  to 
satisfying  this  test,  and  he,  therefore,  agrees  with  me  in 
my  conclusion  that  this  formula  is  the  best,  but  for  very 
different  reasons.  Its  failure  perfectly  to  satisfy  this  test 
is  regarded  by  him  as  a  blemish  or  shortcoming. 

Much  intellectual  labor  has  thus  been  expended  in  a 
vain  effort  to  find  a  formula  which  will  yield  the  absolutely 
consistent  results  required  by  the  circular  test  and  still 
be  satisfactory  in  other  respects. 

The  simple  or  the  constant  weighted  geometric  index 
number  was  favored  by  Jevons  and  Walras  and  several 
later  writers,  including  Flux  and  March,  chiefly,  it  would 
seem,  because  it  satisfies  this  test,  always  giving  self- 
consistent  results  whatever  year-to-year  calculations  are 
made. 


CHAPTER  XIV 

BLENDING  THE  APPARENTLY  INCONSISTENT  RESULTS 

§  1.  Introduction 

I  THINK  most  students  of  index  numbers  would  be 
inclined  to  say  of  the  circular  test  that  theoretically  it 
ought  to  be  fulfilled,  but  that  practically  it  is  not ;  and 
evidence  would  be  cited  from  index  numbers,  like  Formula 
1,  which  have  large  circular  gaps.  We  have  found  in 
Chapter  XIII  that  the  exact  opposite  is  true;  that 
theoretically  the  circular  test  ought  not  to  be  fulfilled, 
but  that  practically  it  is  fulfilled  by  the  best  index  num- 
bers, and  our  evidence  is  the  infinitesimal  gap  worked  out 
for  Formula  353  and  the  other  curves  in  the  "  superlative  " 
group. 

Theoretically,  every  pair  of  years  has  its  own  particular 
index  number  dependent  on  the  prices  and  quantities 
pertaining  to  those  particular  years,  regardless  of  any  other 
year  or  years.  As  a  consequence  of  this  individualism 
of  index  numbers  there  is,  theoretically,  a  lack  of  team 
play,  as  it  were,  between  the  index  numbers  connecting 
different  years  and  there  is,  in  consequence,  an  appearance 
of  mutual  inconsistency.  It  follows  that,  to  secure  the 
theoretically  most  perfect  result,  for  the  sake  of  finding 
the  very  best  for  each  pair  of  years,  we  should,  for  a  given 
series  of  years  and  with  a  given  formula,  work  out  every 
possible  index  number  connecting  every  possible  pair 
of  years  among  all  the  years  considered.  Thus,  for  the 
six  years  taken  for  the  calculations  of  this  book,  we  should, 
theoretically,  work  out  the  index  number 

297 


298         THE  MAKING  OF  INDEX  NUMBERS 

between  1913  as  base  and  each  of  the  other  five  years 

11         1014    "       "       "        il      "     tt        tl        li        il 
"        lc       u         u       t(      t{         lt         l(         (( 


t{        1916   "      u      "        "      "     "       "       "       " 
t(          1Q17    u        ft       tl         ft       u      lt         <(         (C         tl 

u        1918   "      "      "       "      "     "       "       "       " 

That  is,  we  should  use  every  year  as  base  for  all  the  rest. 
This  would  give  us  a  complete  set  of  index  numbers  be- 
tween every  possible  pair  of  years,  each  separate  figure 
having  its  own  special  meaning,  and  to  be  used  only  for 
the  one  comparison,  i.e.  between  the  two  years  for  which 
it  is  calculated. 

This  would  make  30  separate  index  numbers.  In  this 
list  of  30,  every  pair  of  years  enters  twice,  in  opposite 
directions  ;  once  when  one  of  the  two  years  is  the  base 
and  again  when  the  other  is  the  base.  Thus  there  are 
only  15  pairs  of  years,  each  compared  through  two  index 
numbers,  which  are  reciprocals  when  Test  1  is  met.  Of 
these  15,  we  have,  as  the  reader  will  remember,  actually 
worked  out  index  numbers  for  nine  by  each  of  our  134 
formulae,  namely,  the  five  on  1913  as  base,  which  consti- 
tute the  "  fixed  base"  series  ;  and  the  five  which  constitute 
the  "  chain  "  system,1  less  one  duplication,  inasmuch  as 
the  first  figure  (that  for  1914)  is  common  to  both  the 
fixed  base  and  chain  systems.  The  other  six,  not  worked 
out,  are  those  connecting  years  1914  and  1916,  1914 
and  1917,  1914  and  1918,  1915  and  1917,  1915  and  1918, 
1916  and  1918. 

For  a  series  of  ten  years,  there  would  be,  instead  of  15 

10  X  9 
such  "  permutations,"  —  -~  —  ,  or  45  separate  index  num- 

2 

bers,  of  which  nine  (connecting  1913  with  each  of  the  nine 

1  The  complete  fixed  base  series  and  some  of  the  chain  series  for  all  the 
134  formulae  are  given,  as  previously  noted,  in  Appendix  VII. 


BLENDING  THE  INCONSISTENT  RESULTS    299 

other  years)   would  be  the  ordinary  fixed  base  series  and 
eight  others  would  be  added  in  the  "  chain."     For  20 

20  X  19 
years  there  would  be  — £ — ,  or  190  separate  index  num- 

2i 

bers.    For  100  years  there  would  be  10°  *  ",    or    4950 

2 

separate  index  numbers. 

To  calculate  such  an  enormous  quantity  of  separate 
index  numbers,  for  the  sake  of  finding  the  very  best  for 
each  pair  of  years,  and  to  do  so  every  time  we  are  con- 
fronted with  the  problem  of  tracing  price  movements 
through  a  series  of  years,  would  clearly  entail  very  great 
labor  and  expense.  Would  it  be  worth  while?  If  not, 
that  is,  if,  in  practice,  we  must  forego  a  theoretically 
perfect  set  of  index  numbers  for  every  possible  pan*  of 
years,  what  will  be  the  best  course  to  pursue  from  a 
practical  point  of  view?  Shall  we  content  ourselves  with 
the  fixed  base  set  and  use  that  series,  not  only  for  its  proper 
purpose  of  comparing  the  fixed  base  year  with  each  other 
year,  but  also  for  the  theoretically  improper  purpose  of 
comparing  any  other  two  years?  If  so,  shall  we  use  the 
first  year  as  the  base  from  which  to  make  our  once-for-all 
set  of  computations,  or  shall  we,  for  base,  adopt  an  average 
covering  several  years?  Or  shall  we  employ  the  chain 
system  which  is  theoretically  proper  only  for  comparing 
any  two  successive  years  but  improper  for  comparing  any 
other  two  years?  Or  shall  we  use  both  the  fixed  base 
and  chain  systems?  We  are  now  ready  to  work  out 
answers  to  these  questions. 

§  2.  Formula  353  Calculated  on  Each  Separate  Year 

as  Base 

To  illustrate  these  problems,  if  we  take  353  as  bur  formula  and  1913 
as  base,  we  get  the  following  results :  for  1916,  114.21,  and  for  1918,  177.65. 
But,  theoretically,  this  does  not  justify  us  in  assuming  that  the  price  levels 


300         THE  MAKING  OF  INDEX  NUMBERS 

of  1916  and  1918,  compared  directly  and  properly  with  each  other,  stand 
as  114.21  to  177.65.  Again,  the  chain  system  gives  correctly  the  com- 
parison only  between  two  consecutive  years.  Thus,  it  tells  us  that  the 
price  levels  of  1916  and  1917  stood  in  the  ratio  of  114.32  and  162.23  and 
that  the  levels  of  1917  and  1918  stood  in  the  ratio  of  162.23  and  178.49. 
But  theoretically  these  do  not  justify  us  in  assuming  that  the  price  levels 
of  1916  and  1918  stand  in  the  ratio  of  114.32  and  178.49.  The 
theoretically  correct  comparison  between  1916  and  1918  must  be  made, 
neither  by  reference  to  the  first  year,  1913,  nor  by  reference  to  the  inter- 
mediate year,  1917,  but  directly.  That  is,  either  1916  must  be  the  base  and 
1918  calculated  from  it,  or  vice  versa. 

By  such  direct  comparison,  taking  1916  as  the  base  and  calling  it,  not 
100  but  114.32  (to  facilitate  comparison  with  the  above  figures),  we  find 
that  prices  actually  rose  between  1916  and  1918  in  the  ratio  of  114.32  to 
178.36  instead  of,  as  per  the  chain  series,  from  114.32  to  178.49  or,  as 
per  the  fixed  base  (1913)  series,  from  114.21  to  177.65. 

Table  40  gives  the  complete  set  of  index  numbers  for  the  years 
1913-1918  with  each  year  as  base.  The  first  line  gives  the  index  numbers 
with  1913  as  the  fixed  base,  taken  as  100  per  cent,  as  usual.  In  this  series, 
the  index  number  for  1914  is,  for  instance,  100.12.  The  next  line  gives 
the  index  numbers  with  1914  as  base,  taken  not  as  100  but,  to  facilitate 
comparisons,  as  100.12  (as  in  the  line  above).  Thus,  with  1914  as  such 
a  base,  1915  is  100.23.  The  third  line  gives  the  index  numbers  with  1915 
as  base  taken  (from  the  line  above)  as  100.23;  for  instance,  with  1915 
as  such  a  base  1916  is  114.32,  and  so  on,  each  successive  year  being  thus 
taken  as  base  but  not  as  100  (excepting  1913). 

The  figures  mentioned  as  base  figures  are  italicized  in  a  diagonal 
and  they  themselves  constitute  the  chain  figures.  That  is,  the  diagonal 
series  is  the  chain  series.  By  this  device,  for  example,  the  right  and  bottom 
corner  figure,  178.49,  serves  the  double  purpose  of  being  at  once  in  the 
chain  and  in  the  1918  fixed  base  series  just  as  the  diagonally  opposite 
(left  upper)  corner  figure  (100.00)  serves  the  corresponding  double  purpose 
of  being  at  once  the  beginning  of  the  chain  and  of  the  1913  fixed  base 
series.  In  the  same  way,  the  second  row  of  figures  is  the  fixed  base  series 
where  1914  is  the  base,  and  is  taken  not  as  100,  but  as  100.12,  the  chain 
figure.  Thus  all  figures  in  the  diagonal  serve  as  the  base  for  all  the  years 
on  the  same  line  as  well  as  a  link  in  the  chain  (the  diagonal). 

If  such  a  table  were  to  be  used  in  practice  it  would  be  used  as  follows. 
The  first  line,  or  ordinary  fixed  base  figures  (1913  being  the  base),  would 
be  used  only  for  comparing  any  given  year  such  as,  say,  1917  with  this  base, 
1913,  and  not  for  comparing  it  (1917)  with  any  other  year  such  as,  say, 
1915.  If  we  wished  to  compare  1917  with  1915  we  should  find  in  the 
table  the  line  in  which  one  of  these  two  years  is  the  base  (an  italicized 
figure),  for  instance,  the  third  line.  There  1915  is  the  base,  and  is  taken  as 
100.23.  On  this  base,  1917  is  found  to  be  161.86.  Consequently,  the 
best  measure  for  the  rise  of  prices  between  1915  and  1917  is  this  rise  from 
100.23  to  161.86.  It  is,  strictly,  not  the  rise  given  in  the  first  line  in  the 
table,  by  the  ordinary  fixed  base  system.  It  is  there  represented  as  a  rise 
from  99.89  to  161.56  although  in  this  case  the  two  comparisons  differ 
almost  inappreciably. 


BLENDING  THE  INCONSISTENT  RESULTS    301 


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302         THE  MAKING  OF  INDEX  NUMBERS 

In  the  above  comparison  1915  was  taken  as  the  base  year  and  1917  as 
the  given  year.  We  could,  of  course,  reverse  the  bases,  taking  the  fifth 
line  where  1917  is  162.23,  for  base,  in  which  case  the  given  year  1915  is 
100.46,  thus  giving  the  rise  of  prices  between  1915  and  1917  as  100.46  to 
162.23;  this  comparison  is,  of  course,  exactly  the  same  as  the  first  (i.e. 
100.23  :  161.86 :  :  100.46  :  162.23)  because,  as  we  know,  our  formula  (353) 
satisfies  the  time  reversal  test. 


§  3.  The  Differences  Due  to  Differences  of  Base  are 

Trifling 

By  Table  40  we  may  very  readily  see  the  trifling  effects  of  shifting  the 
base  from  one  year  to  another.  For  1913  the  figures  (for  prices)  in  the  left- 
most vertical  column  vary  only  from  100  to  100.47;  for  1914,  from  100.12 
to  100.76;  for  1915,  from  99.89  to  100.46;  for  1916,  from  114.11  to  114.41 ; 
for  1917,  from  161.21  to  162.23;  for  1918,  from  177.65  to  178.80.  These, 
which  are  the  extreme  discrepancies  brought  about  for  each  year  by  shifting 
the  base  each  year,  range  only  from  one  third  of  one  per  cent  to  two  thirds 
of  one  per  cent ! 

Let  us  take  the  last  and  largest  of  these  and  state  the  meaning  of  the 
discrepancy.  It  is  the  discrepancy  between,  on  the  one  hand,  177.65  as 
the  index  number  for  1918  on  the  base  1913  taken  as  100  per  cent,  and, 
on  the  other  hand,  178.80  for  1918  on  the  base  1915  taken  as  100.23. 
And,  to  proceed  back  to  1913,  this  last  named  figure  on  the  diagonal, 
100.23,  was  found  as  the  index  number  for  1915  on  1914  as  base  taken  as 
100.12  (preceding  line),  which,  in  turn  (next  preceding  line),  was  found  as 
the  index  number  for  1914  on  1913  as  base  taken  as  100.00.  In  other 
words,  by  the  true  direct  comparison,  taking  1913  as  100  per  cent,  we 
find  that  the  index  number  of  1918  is  177.65  per  cent;  but  by  the  indi- 
rect comparison,  starting  with  the  same  base  and  proceeding  one  link  to 
1914  (diagonal),  thence  another  link  (diagonal)  to  1915,  and  then  jump- 
ing (level)  to  1918,  we  get,  not  177.65,  but  178.80,  or  two  thirds  of  one 
per  cent  more. 

Thus  the  difference  between  the  various  barometers  of  price-and-quan- 
tity-changes  given  in  the  table  are  trifling.  Nevertheless,  it  is  interesting 
to  note  that,  as  between  1914  and  1915  where  the  two  index  numbers  are 
virtually  equal,  there  is  enough  difference  to  tip  the  scales  from  one  direc- 
tion to  the  other.  According  to  the  first  line,  or  ordinary  fixed  base  system, 
1913  being  the  base,  the  price  level  seems  to  fall  between  1914  and  1915 
(from  100.12  to  99.89,  or  a  quarter  of  one  per  cent)  and  a  slight  fall  between 
the  same  years  (1914  and  1915)  is  likewise  indicated  in  the  last  three  lines, 
i.e.  with  1916,  1917,  or  1918  as  base;  whereas  by  the  direct,  or  true,  com- 
parison between  1914  and  1915,  i.e.  with  1914  as  base  or  1915  as  base  (see 
second  line  and  third  line),  we  note  that  the  price  level  is  found  to  rise 
from  100.12  to  100.23,  or  one  ninth  of  one  per  cent. 

The  reader  will  notice  that  each  italicized  chain  figure  (say  for 
1915)  is  duplicated  immediately  above  and  also  immediately  below:  — 
above,  because  the  italicized  1915  figure  was  purposely  taken  from 


BLENDING  THE  INCONSISTENT  RESULTS    303 

the  line  above  to  start  off  the  calculations  on  the  new  1915  base;  and 
below  (the  1916  line)  because  the  1915  year  is  there  calculated  backward 
from  the  1916  base  by  a  formula  which  complies  with  Test  1.  In  a  word, 
in  the  1915  line,  1916  is  calculated  from  1915;  and  in  the  1916  line  1915  is 
calculated  from  1916,  with  a  formula  which  works  both  ways,  i.e.  com- 
plies with  Test  1. 

Graphically,  Chart  56,  plotting  Table  40,  shows  the 
results  of  applying  Formulae  53  and  54  and  their  cross,  353, 
on  each  of  the  six  bases.  The  upper  three  sets  give  these 
18  curves  (six  for  each  formula)  individually,  separated  by 
spaces,  while  the  lower  three  give  a  composite  of  each 
set. 

It  is  clear  that  the  differences  are  extremely  trifling, 
and,  for  353,  scarcely  perceptible.  The  preceding  table 
and  chart  thus  show  in  another  way  what  we  saw  in  the 
last  chapter  specifically  by  means  of  the  circular  test, 
namely,  how  remarkably  little  difference  it  makes  what 
the  base  or  bases  may  be  from  which  we  calculate  Formula 
353. 

In  view  of  this  virtual  agreement  between  the  curves, 
whatever  year  is  taken  as  the  base,  it  is  perfectly  clear 
that  for  Formula  353  (and  the  same  would  be  true  of  any 
other  good  formula)  it  would  be  a  waste  of  time,  in  the 
practical  calculation  of  index  numbers,  always  to  cal- 
culate all  possible  inter-year  indexes.  Any  one  series 
will  suffice. 

In  short,  while  theoretically  the  circular  test  ought  not 
to  be  fulfilled,  and  shifting  the  base  ought  to  yield 
inconsistencies,  the  inconsistencies  yielded  are  so  slight 
as  practically  to  be  negligible.  To  use  for  each  formula 
all  the  six  curves  (for  six  years  —  more,  for  more  years) 
would  only  multiply  the  time,  labor,  and  expense  by  a 
large  factor,  without  serving  any  useful  purpose.  In 
fact,  it  would  be  a  positive  nuisance.  A  single  curve 
will  suffice  for  all  practical  purposes. 


304 


THE  MAKING  OF  INDEX  NUMBERS 


§  4.  Index  Numbers  on  Different  Bases  may 
well  be  Blended 

Every  one  of  the  six  curves  is  strictly  correct  only 
for  the  limited  comparison  for  which  it  is  constructed. 


Comparison  For  Six  Bases 
of  Formulae  5%  54, 353 
(Prices) 


'15 


'16 


'17 


CHABT  56P.  These  curves,  especially  the  three  lower,  which  are  mere 
composites  of  those  above  (i.e.  found  by  plotting  all  on  the  same  scale,  in- 
stead of  separating  them  as  above),  indicate  that  the  differences  resulting 
from  a  shift  of  base  are  least  for  353,  but  comparatively  slight  for  53  and 
54  also. 

There  remains  the  practical  question :  if  we  are  not  going 
to  use  all  six,  what  single  curve  is  the  best  one  to  use 
in  their  place,  for  the  general  purpose  of  all  com- 


BLENDING  THE  INCONSISTENT  RESULTS    305 

i  _  ( 

parisons  over  a  series  of  years  ?  Doubtless  the  very  best 
as  to  accuracy,  were  it  practicable,  is  the  blend  or  average 
of  all  six.  This  blend  constitutes  Formula  7053,  if  it  can 
be  dignified  by  the  name  of  formula.  It  is,  of  course, 
merely  an  average  of  the  six  sets  of  particular  figures  de- 
rived by  Formula  353.  This  is  a  compromise  single  series 

Comparison  For  Six  Bases 
of  Formulae  53.54.353 
(Quantities) 


'14  '15  16  '17 

CHAET  56Q.    Analogous  to  Chart  56P. 


'IQ 


of  six  figures  that  can  be  substituted  for  the  whole  table 
of  figures,  for  the  purpose  of  blending  all  separate  exact 
comparisons  into  one  general  nearly  exact  comparison. 
With  reference  to  these  averages,  no  figure  in  the  table 
deviates  by  as  much  as  one  half  of  one  per  cent.  The 
"  probable  error  "  of  any  figure  (for  price  indexes,  for 
1917)  is  two  tenths  of  one  per  cent,  and,  for  the  other 
years,  less.  In  other  words,  it  is  just  as  likely  as  not  that 


306         THE  MAKING  OF  INDEX  NUMBERS 

any  figures  of  Table  40  for  1917  taken  at  random  will 
differ  from  the  mean  (or  Formula  7053)  figure  for  1917 
(viz.,  161.53)  by  less  than  two  tenths  of  one  per  cent.1 

This  blend  may  be  compared  to  the  "  chromatic  " 
scale  on  the  piano.  This  chromatic  scale  is  found  by 
"  tempering  "  the  "  natural  "  scale.  By  the  "  natural  " 
scale  a  piano  would  have  but  one  key;  to  obtain  other 
keys  would  require  a  separate  piano  for  each,  all  out  of 
tune  with  one  another.  These  are  blended  into  one  by 
the  chromatic  scale  by  slight  readjustments  of  the  various 
notes.  These  adjustments  change  the  number  of  vibra- 
tions in  the  natural  scale  in  one  case  by  as  much  as  1  in  122, 
or  some  ten  times  as  great  an  adjustment  as  we  are  called 
upon  to  make  in  our  present  problem  of  adjusting  index 
numbers.  In  other  words,  the  "  tempering  "  of  the  piano 
or  "  chromatic  "  scale  relatively  to  the  violin  or  "  natural  " 
scale,  though  imperceptible  to  almost  any  human  ear, 
is  ten  times  as  great  as  the  "  tempering  "  which  is  neces- 
sary to  secure  Formula  7053. 

§  5.  The  Three  Practical  Substitutes  for  Blending 

But  to  calculate  Formula  7053  every  time  we  have  an 
index  number  to  compute  would  require,  first,  calculating 
each  of  the  constituent  curves  and  this,  as  has  been  said, 
could  be  done  only  at  prohibitive  costs.  From  a  practical 
point  of  view,  there  are  only  three  single  curves  worth 
considering:  (1)  that  obtained  by  using  the  first  year 
1913  as  base  (the  ordinary  fixed  base  Formula  353  or  its 
rivals) ;  (2)  that  by  using  the  chain  of  successive  bases 
(also  by  353  or  its  rivals) ;  and  (3)  that  by  using  6053 
(or  its  rival  6023),  which  are  like  53  (or  23),  except  that 

1  Formula  7053,  as  here  used,  begins  in  1913  with  100.22.  For  conven- 
ience, we  may  reduce  this  to  100  and  reduce  all  the  figures  for  all  the  other 
years  accordingly.  Both  forms  are  given  in  the  preceding  table,  but 
only  the  last  named  in  Appendix  VII. 


BLENDING  THE  INCONSISTENT  RESULTS    307 


the  base  is  not  a  single  year  but  an  average  formed  from 
several  or  all  the  years  concerned.  Such  a  formula  may 
be  called  aggregative  (or  geometric)  formula  weighted  I  with 
broadened  base.  One  of  its  chief  claims  to  consideration 
is  that  it  requires  fewer  statistical  data  to  be  furnished 
than  does  353. 

To  determine  which  of  these  three  (353  fixed  base, 
353  chain,  or  6053  broadened  base)  is  the  most  accurate, 

TABLE  41.     FOUR  SINGLE  SERIES  OF  SIX  INDEX  NUMBERS 
AS  MAKESHIFTS  FOR  THE  COMPLETE  SET  OF  TABLE  40 

(Prices) 


1913 

1914 

1915 

1916 

1917 

1918 

Formula  6053  (broadened 
base,  1913-1918) 
Formula  353  (fixed  base, 
1913) 
Formula  353  (chain) 

100. 
100. 
100. 

99.79 
100.12 
100.12 

99.85 
99.89 
100.23 

114.04 
114.21 
114.32 

161.59 
161.56 
162.23 

177.88 
177.65 
178.49 

Formula  7053  (blend) 

100. 

100.09 

99.96 

114.03 

161.53 

177.90 

This  table  shows  that  the  chain  system  is  the  most  erratic  of  the  three 
as  compared  with  Formula  7053  and  that  there  is  practically  no  choice 
between  the  other  two. 

The  figures  for  quantities  show  the  same  result. 

TABLE  42.    FOUR  SINGLE  SERIES  OF  SIX  INDEX  NUMBERS 
AS  MAKESHIFTS  FOR  THE  COMPLETE  SET  OF   TABLE  40 

(Quantities) 


1913 

1914 

1915 

1916 

1917 

1818 

Formula  6053  (broadened 
base,  1913-1918) 
Formula  353  (fixed  base, 
1913) 
Formula  353  (chain) 

100. 
100. 
100. 

99.00 
99.33 
99.33 

108.91 
109.10 
108.72 

119.13 
118.85 
118.74 

118.99 
118.98 
118.49 

125.16 
125.37 
124.77 

Formula  7053  (blend) 

100. 

99.37 

109.02 

119.04 

119.00 

125.20 

308 


THE  MAKING  OF  INDEX  NUMBERS 


it  is  only  necessary  to  ascertain  which  of  them  is  nearest 
to  the  best  blend,  namely,  7053. 

Numerically,  Tables  41  and  42  on  page  307  give  these 
three  sets  of  figures  and  also  the  theoretically  best  blend, 
Formula  7053,  for  comparison. 


Optional  Varieties  of 
(Prices) 


6053  CIS-IS) 
353  Fixed  base 
35Scho,n 

7053 


'15 


16 


77 


'IS 


73  Mx 

CHART  57P.  The  agreement  between  the  broadened  base  index  num- 
ber (6053),  the  blend  of  the  six  curves  of  353  (7053),  and  353  itself  (whether 
with  1913  as  a  fixed  base  or  with  the  chain  system),  is  so  close  that,  were 
precision  the  only  consideration,  there  would  be  almost  no  choice  between 
these  four. 

Graphically,  Chart  57  gives  these  three  curves  and  also 
the  theoretically  best  formula,  7053.  They  are  absolutely 
indistinguishable  to  the  eye. 

Our  conclusion  is,  then,  that  either  Formula  353,  fixed 
base  1913,  or  Formula  6053,  broadened  base  1913-1918,  is 
the  best  compromise  on  the  score  of  accuracy.  On  the 
score  of  other  and  more  practical  considerations,  such  as 
speed  of  computation,  more  will  be  said  in  a  later  chapter. 

§  6.   Chain  vs.  Fixed  Base  System 

The  chain  system  is  of  little  or  no  real  use.  The  chief  arguments  in 
favor  of  the  chain  system  are  three  :  (1)  that  it  affords  more  exact  com- 


BLENDING  THE  INCONSISTENT  RESULTS    309 


parisons  than  the  fixed  base  system  between  the  current  year  and  the 
years  immediately  preceding  in  which  we  are  presumably  more  interested 
than  in  ancient  history;  (2)  that,  graphically,  the  year-to-year  lines  of 
the  price  curve  have  the  correct  current  directions,  whereas  in  the  fixed 
base  system  the  year-to-year  lines  are  slightly  misleading,  merely  connect- 
ing points  each  of  which  is  really  located  relatively  to  the  base  or  origin  only, 
and  not  to  its  neighbors ;  and  (3)  that  it  makes  less  complicated  the  neces- 
sary withdrawal,  or  entry,  or  substitution  of  commodities,  as  time  and 
change  constantly  require. 

As  to  the  first  argument,  though  I  have  myself  used  it  in  the  past,  I 
have  come  to  a  lower  estimation  of  its  importance ;  partly  (and  chiefly) 
because  the  present  investigation  has  shown  that,  in  the  case  of  all  good 
index  numbers,  there  is  no  really  perceptible  difference  between  the  chain 

Optional   Varieties   of  353 
(Quantities) 


\S* 


15  16  17 

CHART  57Q.    Analogous  to  Chart  57P. 


78> 


and  the  fixed  base  figures;  partly  because,  for  years  to  come,  we  shall 
be  interested  in  comparisons  with  antecedent  and  pre-war  years  quite  as 
much  as  with  the  immediately  preceding  years ;  and  partly  because  I  have 
come  to  realize  that  the  ordinary  user  of  index  numbers  uses  chiefly  not  the 
diagram  but  the  numerical  figure,  and  he  thinks  of  this  figure  as  relative  to 
the  base.  Therefore,  it  is  better  that  it  should  accurately  express  the  rela- 
tion to  the  base.  This  the  fixed  base  figure  does. 

The  second  argument  —  the  one  concerning  graphic  representation  — 
is  sufficiently  answered  by  the  fact  that  the  eye  is  not  accurate  enough  to 
distinguish  between  the  fixed  base  and  chain  base  curves  given  by  any  of 
the  better  formulae.  Very  minute  differences  can  be  perceived  only  by 
printed  figures. 

Theoretically,  it  may  be  said  that  the  graphic  curve  for  the  fixed  base 
system  is  an  anomaly.  To  represent  the  fixed  base  and  chain  curves  most 
appropriately,  we  ought  to  draw  only  the  chain  curve  from  year  to  year, 
i.e.  from  ball  to  ball,  whereas,  when  we  use  the  fixed  base  points,  we  ought 
to  connect  these,  not  with  each  other,  but  each  directly  with  the  base  point 
or  origin. 

In  Chart  58  (fixed  base,  using  the  simple  median)  the  connecting  lines 
between  each  point  and  the  origin  are  graphically  indicated  (dark  short 
lines  drawn  only  part  way  toward  the  origin  to  avoid  confusing  the  eye) ; 
but  these  would  not  give  much  help  to  the  onlooker  were  not  their  ends 


310         THE  MAKING  OF  INDEX  NUMBERS 

connected  by  the  dotted  curve  after  the  usual  fashion  of  the  fixed  base 
curves. 

§7.   Splicing 

The  strongest  argument  for  the  chain  system  is  the  third,  i.e.  the  im- 
munity it  gives  from  any  complications  arising  out  of  the  withdrawal  of 
any  commodity  from  the  index  number,  or  the  entry  of  a  new  commodity, 
or  both  at  once,  i.e.  the  substitution  of  a  new  for  an  old. 

Simple   Median  .x 

and  Quartiles    drawn  from  origin         / 
(Prices) 


75  74  15  76  17  76 

CHART  58.  Showing  how,  strictly  speaking,  the  fixed  base  index  num- 
bers should  be  represented  —  by  lines  radiating  from  the  fixed  base  to  the 
given  years.  The  lines  are  for  the  median  (in  the  center),  those  above  and 
below  representing  the  quartiles.  The  dotted  connecting  line  is  needed  to 
help  the  eye  despite  the  fact  that,  strictly  speaking,  its  directions  do  not 
represent  year-to-year  index  numbers. 

It  often  happens  that  we  wish  to  drop  some  commodity  from  the  list 
because  of  its  ceasing  to  be  quoted,  or  of  its  becoming  obsolete  or  super- 
seded. And,  likewise,  it  often  happens  that  we  wish  to  include  a  new 
commodity  because  of  a  new  invention  or  a  change  in  customs.  Still 
of  tener  must  substitutions  be  made  by  replacing  one  grade  or  style  of  goods 
by  another.  When  the  chain  system  is  used  these  operations  create  no 
embarrassment,  no  matter  what  formula  is  used;  for,  under  this  system, 


BLENDING  THE  INCONSISTENT  RESULTS    311 

a  new  start  is  made  each  year  and  the  next  link  can  be  forged  independently 
of  all  those  preceding. 

But  under  the  fixed  base  system  these  changes  usually  make  Gordian 
knots  to  cut.  In  some  cases  there  is  no  difficulty.  Thus,  if  we  drop  one 
brand  of,  say,  condensed  milk  and  substitute  another  and  if  the  newly 
marketed  brand  has,  at  the  time  of  the  change,  the  same  price  as  the  old, 
it  may  be  substituted  without  any  jar  or  adjustment,  even  though  it  did 
not  exist  in  the  base  year.  Similarly,  if  one  grade,  say,  of  wheat  which  did 
exist  in  the  base  year  but  was  not  used  in  the  index  number,  is  now  sub- 
stituted for  another  and,  though  their  prices  per  bushel  do  differ,  their 
price  relatives  do  not  differ  in  terms  of  their  base  year  prices,  we  may  readily 
make  the  transference.  Again,  if  the  withdrawal  or  entry  does  not  change 
the  index  number,  there  is  no  trouble.  This  supposition  implies,  of  course, 
in  the  case  of  entry,  that  the  newly  entered  commodity  was  also  quoted 
in  the  base  year.  But  in  all  other  cases  under  the  fixed  base  system  we 
must  make  some  sort  of  adjustment. 

Let  us  assume  that  the  change  (whether  withdrawal,  entry,  or  substitu- 
tion) changes  the  index  number,  at  the  time,  from  150  under  the  old  way  to 
153  under  the  new,  or  by  two  per  cent.  The  new  figure  being  two  per 
cent  above  the  old,  all  future  figures  calculated  by  the  new  way  may  be 
presumed  to  be  two  per  cent  too  high.  Consequently,  what  is  needed  is, 
henceforth,  after  calculating  by  the  new  way,  to  trim  down  the  result 
by  that  much.  That  is,  beginning  with  the  153,  every  index  number  after 
being  duly  calculated  is  to  be  reduced  in  the  ratio  of  153  to  150. 

But  in  cases  where  an  entirely  new  commodity  enters,  so  that  no  base 
year  quotations  exist,  we  cannot  enter  it  at  all,  in  the  fixed  base  system, 
on  all  fours  with  the  rest.  If  it  is  a  case  of  substitution  for  a  commodity 
to  be  withdrawn,  we  may  splice  it  on  to  the  old  series  of  quotations  for 
the  withdrawn  commodity.  Thus,  if  the  old  commodity,  at  the  time  of 
withdrawal,  stood  at  120,  the  new  may  be  arbitrarily  entered  in  its  place  as 
120  (despite  the  fact  that  there  was  no  100  for  it  in  the  first  place)  and  its 
future  price  relatives  computed  in  proportion.  If  the  new  commodity  is 
not  to  be  substituted  for  an  old,  but  added  as  one  more  on  the  list,  we  may 
arbitrarily  give  as  its  price  relative  at  the  time  a  figure  equal  to  the  index 
number  itself.  That  is,  if  the  index  number  at  the  time  is  130,  the  new 
commodity  may  start  off  with  130  as  its  price  relative  (despite  the  fact 
that  there  never  was  any  100  for  it). 

In  short,  the  fixed  base  system  is  objectionable  because  it  sometimes 
requires  patching.  The  chain  system  never  does.  But  this  objection  to 
the  fixed  base  system  is  not  very  serious.  Besides,  the  patching  may  be 
largely  or  wholly  avoided  if,  as  indicated  in  a  later  chapter,  we  take  a 
new  start,  not  every  year,  but,  say,  every  decade. 

The  above  explanation  is  stated  in  terms  of  price  relatives  and  applies 
to  all  index  numbers,  except  aggregatives.  To  these  an  analogous  method 
applies.1 

On  the  whole,  therefore,  the  fixed  base  system  (at  least  as  applied 
to  Formula  353)  is  slightly  to  be  preferred  to  the  chain,  because, 


See  Appendix  I  (Note  to  Chapter  XIV,  §  7). 


312          THE  MAKING  OF  INDEX  NUMBERS 

(1)  it  is  simpler  to  conceive  and  to  calculate,  and  means  something 
clear  and  definite  to  everybody ; 

(2)  it  has  no  cumulative  error  as  does  the  chain  system  (as  is  shown  by 
comparison  with  Formula  7053) ; 

(3)  graphically  it  is  indistinguishable  from  the  chain  system. 


§  8.  Broadening  the  Fixed  Base 

We  have  considered  two  of  the  three  series  originally 
contrasted,  viz.,  Formula  353  in  the  fixed  base  and  chain 
systems,  and  between  these  two  we  choose  the  fixed  base 
system.  We  have  also  found  that  in  the  fixed  base  system 
we  can  always  "  patch  "  when  commodities  are  changed 
in  the  formula.  We  have  still  to  consider  the  broadened 
base  system  (which  also  requires  revision  from  time  to 
time)  as  compared  with  the  fixed  base  year  system.  This 
is  easier  to  calculate  than  the  blend  Formula  7053,  and 
distributes  in  a  simpler  way  the  discrepancies  due  to  differ- 
ing bases.  Moreover  it  does  not  require  that  the  cal- 
culator have  at  hand  all  the  yearly  data  needed  for  353. 
He  may  make  his  base  as  broad  as  the  data  available,  or, 
as  may  be  necessary  to  yield  a  good  compromise. 

Broadening  the  base  from  one  year  to  several  requires : 
(1)  taking  as  each  base  price,  not  one  year's  price,  but  an 
average  of  several ;  and  (2)  likewise  taking  as  each  base 
weight  not  one  year's  but  an  average  of  several.1  As 
stated,  the  system  of  weighting  is  analogous  to  system  7. 
It  is  the  same  throughout  the  calculation,  i.e.  constant 
weights  are  used  for  the  entire  series.  For  quantity  in- 
dexes, of  course,  the  analogous  operations  apply. 

We  shall  consider  the  advantages  of  broadening  the 
base  as  applied  to  certain  types  of  formulae.  First,  we 
shall  consider  Formula  6053.  It  is  Formula  53,  except 

1  It  may  be  worth  noting,  however,  that  (1)  is  a  superfluous  procedure  in 
the  cases  of  Formulae  6023  and  6053,  the  results  being  identical  (except  for 
a  constant)  whether  one  year's  price  or  an  average  of  several  is  used. 


BLENDING  THE  INCONSISTENT  RESULTS    313 

that  the  base  values  or  quantities  are  taken  as  the  average 
of  the  values  or  quantities  for  several  years  instead  of 
one.1 

It  seems  to  show  no  real  superiority  over  53.  The 
ranking  of  all  index  numbers  in  Table  28  shows  For- 
mula 53  actually  closer  to  353  than  is  6053  (1913-1918), 
the  six  years  indicated  being  the  broadened  base,  their 
average  of  prices  being  the  base  prices  in  place  of  the 
Po's  of  53,  and  their  average  quantities  being  the  weights 
in  place  of  the  go's  of  53.  Again,  it  shows  Formula  53 
nearly  as  close  to  353  as  6053  (1913-1916),  and  not  much 
less  close  than  is  6053  (1913-1914). 2 

So  far  as  the  aggregative  type  is  concerned,  therefore, 
Formula  53  seems  about  as  good  a  substitute  for  7053 
as  6053,  and,  of  course,  it  is  easier  to  compute.  If  the 
broadened  base  Formula  6053  has  any  advantage  over  53, 
that  advantage  is  too  small  to  show  itself  in  the  cases  here 
available,  including  those  for  prices  and  quantities  of 
the  12  crops,  and  for  prices  and  quantities  of  stocks  on 
the  Stock  Exchange  given  in  Chapter  XI. 

We  may,  therefore,  conclude  with  reasonable  safety 
that  Formula  53  is  always  a  good  makeshift  for  the  ideal 
formula,  353,  or  for  the  ideal  blend,  7053.  Broadening 
the  base  to  make  6053  seems  a  superfluous  procedure.3 

1  This  derivative  of  Formula  53  by  broadening  the  base  is,  of  course,  the 
same  as  that  derived  from  Formula  3  by  broadening  the  base.     So  derived 
it  might  be  called  6003. 

2  The  above  comparisons  were  made  with  Formula  353  fixed  base  as 
the  standard  of  comparison,  but  if  Formula  7053  be  used  instead,  we  get 
the  same  results. 

3  The  only  case  where  there  might  be  any  really  perceptible  advantage 
in  Formula  6053  over  Formula  53  is  in  such  a  case  as  that  of  the  12  crops 
used  by  Persons  and  Day,  i.e.  where  there  is  a  large  correlation  between 
the  price  relatives  and  the  quantity  relatives  so  that  Formula  53  has  a 
slight  bias,  second  hand,  as  it  were.     But  even  in  such  a  case  the  advantage 
is  not  large,  as  is  clear  from  the  fact  that  53  and  54  are  so  close  together 
(see  Charts  47  and  48)  and,  therefore,  so  close  to  353. 


314         THE  MAKING  OF  INDEX  NUMBERS 

§  9.  The  Geometric  Formula  Weighted  7  with 
Broadened  Bases 

When  we  turn  from  the  aggregative  type  to  the  geo- 
metric type,  we  find  a  different  situation.  In  this  case  a 
broadening  of  the  base  (Formula  6023)  does  help  ma- 
terially. Professors  Persons  and  Day  of  Harvard  have 
made  much  use  of  Formula  6023.  Because  of  their  ad- 
vocacy I  have  calculated  6023  in  order  to  see  whether 
this  process  of  broadening  the  base  would  reduce  the 

353 **  6023  Compared  /?// 

For  12  Leading  Crops  (Dqy&  Persons) 
(Prices) 


553 
€023 


\5% 


W  VS  '90  95  W  '05  '10  75  20 

CHART  59P.  Showing  the  close  agreement  between  Day's  index  num- 
ber (6023)  and  the  ideal  (353)  for  prices  of  12  crops  with  a  consistent  but 
faint  trace  of  downward  bias  in  6023  (1910  is  the  base). 

downward  bias  of  23.  Evidently  it  does ;  for  all  the  three 
forms  of  Formula  6023  which  have  been  calculated  lie, 
in  Table  28,  nearer  353  than  does  23.  This  is  because 
the  price  relatives  on  the  broadened  base  disperse  much 
less  widely  than  do  those  used  in  calculating  Formula 
23  and,  as  we  know,  bias  decreases  rapidly  with  a  de- 
crease of  dispersion.  The  reason  why  broadening  the 
base  makes  so  much  more  improvement  over  Formula  23 
than  over  53  is  that  there  is  more  room  for  improvement ; 
for  23,  on  1913  as  a  base,  has  a  distinct  downward  bias. 


BLENDING  THE  INCONSISTENT  RESULTS    315 

It  belongs  to  group  "  1-  "  in  our  five-tined  fork.  Broad- 
ening the  base  to  include  the  two  years,  1913  and  1914,  re- 
duces this  bias.  Broadening  it  to  include  four  years, 
1913-1916,  reduces  it  still  further.  This  is  shown  in  the 
following  table : 


353  and  6023  Compared 
For  12  Leading  Crops  (Day&  Persons) 

(Quantities) 


'60 


CHART  59Q.    Analogous  to  Chart  59P.    The  downward  bias  of  6023  is 
more  evident.     (1910  base.) 


TABLE  43.     THE   INFLUENCE  OF   BROADENING   THE   BASE 
IN   REDUCING   BIAS 

(Prices) 


FORMULA  No. 

BASE 

1913 

1914 

1915 

1916 

1917 

1918 

23 
6023 
6023 

1913 
(1913-1914) 
(1913-1916) 

100. 
100. 
100. 

99.61 
100.12 
99.93 

98.72 
99.50 
99.88 

111.45 
112.25 
113.61 

154.08 
153.53 
156.61 

173.30 
173.45 
175.32 

177.65 
177.90 

353 
7053 

1913 
(blend) 

100. 
100. 

100.12 
100.09 

99.89 
99.96 

114.21 
114.03 

161.56 
161.53 

But  the  figures  are  still  below  the  standard  (either  353, 
fixed  base,  or  7053)  all  along  the  line.  Several  other 
calculations  harmonize  with  this  conclusion. 


316         THE  MAKING  OF  INDEX  NUMBERS 

After  I  had  made  these  calculations  for  the  36  commodities,  Professor 
Persons  published  his  defense  of  Day's  index  number  (Formula  6023)  .* 
His  calculations,  which  are  for  12  crops,  are  reproduced  in  Charts  59P,  59Q, 
and  60P,  60Q,  and  show  a  remarkably  close  agreement  between  Formulae 
6023  and  353.  At  the  same  time  they  show  a  slight  trace  of  downward 
bias  remaining  in  6023,  and  completely  confirm  the  above  conclusions. 
The  base,  in  these  studies  of  Day  and  Persons,  is  broadened  to  the  five  years 
1909-1913 :  that  is,  the  constant  weights  used,  instead  of  being  the  values 
for  the  one  year,  1910,  as  per  Formula  23  (i.e.  instead  of  p0qQ,  etc.),  were 
the  average  values  for  the  five  years  named. 

553  and  6023  Compared 

For  12  Leading  Crops  (Day&  Persons) 
(Prices) 


\8* 


'10  11  *  »  W  »  »  9  9  » 

CHART  60P.    Analogous  to  Chart  59P.     (1910  base.) 

In  Chart  59 P,  Formula  6023  is  below  353  in  four  cases  —in  1880, 1885, 
1895,  and  1915;  and  above  in  three  cases  —  in  1890,  1905,  and  1920. 
In  Chart  59Q  it  is  below  in  seven  cases  —  in  1880,  1885,  1890,  1900,  1905, 
1915,  and  1920;  and  above  in  only  one  case,  namely,  1895.  In  Chart  60 P 
it  is  below  in  four  cases  —  in  1914,  1915,  1917,  and  1919 ;  and  above  in 
three  cases  —  in  1913,  1916,  and  1918.  In  Chart  60Q  it  is  below  in  six 
cases  — in  1911,  1915,  1916,  1917,  1918,  and  1919;  and  above  in  only 
one  case  —  1912.  In  all  the  years  not  mentioned  353  and  6023  coincide. 

All  told,  Formula  6023  is  below  in  21  cases  and  above  in  eight,  thus 
showing  that  its  innate  downward  bias  has  not  quite  been  suppressed  by 
broadening  the  base.  It  is  also  clear  from  an  examination  of  the  charts 
that,  as  we  proceed  in  either  direction  from  the  base,  1910,  the  downward 
bias  of  6023  asserts  itself  increasingly. 

Thus,  by  including  a  sufficient  number  of  years  —  a  full  assortment 
of  all  the  chief  varieties  met  with  in,  say,  a  complete  "business  cycle" 
we  can  partly  2  eliminate  (for  a  time  at  least)  the  bias  of  Formula  23.  The 
longer  and  more  representative  the  period,  the  more  nearly  will  the  bias 

1  Warren  M.  Persons,  "Fisher's  Formula  for  Index  Numbers,"  Review 
of  Economic  Statistics,  May,  1921,  pp.  103-13. 

2  See  Appendix  I  (Note  to  Chapter  XIV,  §  9). 


BLENDING  THE  INCONSISTENT  RESULTS    317 

be  eliminated.  But  in  using  Formula  6023,  the  corrective  effect  of  broad- 
ening the  base  will  wear  off  and  the  downward  bias  gradually  reappear 
after  a  few  years.  Thus,  by  broadening  the  base  from  1913  to  1913-1918, 
the  dispersion  of  our  36  price  relatives  in  1918  is  reduced  from  45.09  per 
cent  to  20.23  per  cent.  This  results,  as  Table  48  shows,1  in  an  even  greater 
reduction  of  the  bias  —  from  7.01  per  cent  to  1.67  per  cent,  and,  as  has 
just  been  stated,  accounts  for  the  improvement  in  the  index  number  from 
broadening  the  base.  But,  as  we  have  seen  in  Chapter  V,  the  dispersion 
always  tends  to  increase  with  the  lapse  of  time.  Sauerbeck's  index  number 
has  a  broad  base  (1867-77).  Yet  the  dispersion  of  the  price  relatives 
used  by  him  amounted,  in  1920,  to  129  per  cent.  This,  as  noted  later, 
has  given  the  index  number  an  upward  bias  of  36  per  cent.  If  Sauerbeck's 
index  number  had  been  calculated  by  Formula  6023  instead  of  by  For- 
mula 1  (or  6001)  its  bias  today  would  have  been  approximately  as  great  in 
the  opposite  direction  since,  as  is  shown  in  Table  7,  Formulae  1  and  23 

353  and  6023  Compared 
For  12  Leading  Crops  (Day  £  Persons) 
(Quantities) 


W  7/  12  13  14  75  19  '17  '16  19 

CHABT  60Q.    Analogous  to  Chart  60P.     (1910  base.) 

have  about  the  same  joint  errors  (except  in  opposite  directions,  of  course). 
The  Day  index,  if  continued  long  enough,  will  inevitably  deteriorate  in  the 
same  way. 

The  general  conclusion  is  that  broadening  the  base  of  the  weighted 
geometric,  by  which  process  Formula  23  is  converted  into  6023,  partially 
eliminates  the  bias  in  the  weighting  of  23,  but  not  entirely.  Consequently, 
the  aggregatives,  Formulae  6053  and  53,  which  are  virtually  free  from  bias, 
are  probably  slightly  better  makeshifts  for  353  than  is  the  geometric  6023, 
which  has  a  very  distinct  bias. 

§  10.  Averaging  Various  Individual  Quotations  for  One 
and  the  Same  Commodity 

Broadening  the  base  implies  an  average  of  the  data  for  a  series  of  years 
and  so  raises  the  question  of  how  that  average  is  to  be  constructed.  As  a 
matter  of  fact,  I  have  used  the  simple  arithmetic  average.  We  need  not 
discuss  this  at  any  great  length,  inasmuch  as  we  have  found  broadening  the 
base  of  little  or  no  importance. 

1  See  Appendix  I  (Note  to  Chapter  V,  §  11). 


318         THE  MAKING  OF  INDEX  NUMBERS 

Essentially  the  same  problem  enters,  however,  whenever,  as  is  usually 
the  case,  the  data  for  prices  and  quantities  with  which  we  start  are  aver- 
ages instead  of  being  the  original  market  quotations.  Throughout  this 
book  "the  price"  of  any  commodity  or  "the  quantity"  of  it  for  any  one 
year  was  assumed  given.  But  what  is  such  a  price  or  such  a  quantity? 
Sometimes  it  is  a  single  quotation  for  January  1  or  July  1,  but  usually  it 
is  an  average  of  several  quotations  scattered  through  the  year.  The 
question  arises:  On  what  principle  should  this  average  be  constructed? 
The  practical  answer  is  any  kind  of  average  since,  ordinarily,  the  variations 
during  a  year,  so  far,  at  least,  as  prices  are  concerned,  are  too  little  to  make 
any  perceptible  difference  in  the  result,  whatever  kind  of  average  is  used. 
Otherwise,  there  would  be  ground  for  subdividing  the  year  into  quarters 
or  months  until  we  reach  a  small  enough  period  to  be  considered  practically 
a  point.  The  quantities  sold  will,  of  course,  vary  widely.  What  is  needed 
is  their  sum  for  the  year  (which,  of  course,  is  the  same  thing  as  the  simple 
arithmetic  average  of  the  per  annum  rates  for  the  separate  months  or  other 
subdivisions).  In  short,  the  simple  arithmetic  average,  both  of  prices  and 
of  quantities,  may  be  used.  Or,  if  it  is  worth  while  to  put  any  finer  point 
on  it,  we  may  take  the  weighted  arithmetic  average  for  the  prices,  the 
weights  being  the  quantities  sold. 

This  problem  of  averaging  the  individual  price  quotations  of  one  in- 
dividual commodity  in  order  to  obtain  "the  price"  for  it  for  the  year  is, 
of  course,  quite  different  from,  and  much  simpler  than  the  main  problem 
of  this  book,  which  is  the  problem  of  constructing  index  numbers  from 
such  yearly  figures  for  many  commodities  after  they  are  individually 
obtained  to  start  with. 

§  11.   Conclusions 

It  appears  that  broadening  the  base  to  secure  a  blend 
is  always  disappointing.  In  the  case  of  the  aggregative 
it  seems  superfluous ;  for  we  cannot  find  that,  in  practice, 
it  is  any  improvement  over  Formula  53.  Moreover  a 
blend  is  a  blur  and  disappoints  our  natural  desire  for 
definiteness.  It  is  neither  flesh,  fish,  nor  fowl.  In  the 
case  of  the  geometric  it  fails  to  suppress  completely  all 
traces  of  weight  bias. 

The  chief  conclusions  of  this  chapter  and  the  last  are : 

1.  Theoretically,   a  complete  set   of    index  numbers 
among  a  number  of  years  consists  of  all  the  possible  index 
numbers  between  every  pair  of  years,  using  Formula  353 
or  any  of  its  peers. 

2.  Practically,   the  apparent  inconsistencies  between 


BLENDING  THE  INCONSISTENT  RESULTS    319 

these  index  numbers  coupling  every  pair  of  years  is  negli- 
gible so  that  the  calculation  of  so  many  would  be  a  waste 
of  time,  effort,  and  money. 

3.  Even  were  such  multiple  calculations  practicable,  — 
connecting  every  possible  pair  of  years  —  they  would 
not  be  helpful  but  confusing,  like  the  conflicting  natural 
scales  in  music.     We  would  be  inclined  to  "  temper  " 
or  "  blend  "  them  into  a  single  series.     The  ideally  best 
blend  would  probably  be  an  average  (Formula  7053)  of  the 
index  numbers  formed  by  calculating  353  on  all  possible 
bases. 

4.  Practically  (and  so  barring  blends,  like  Formula  7053, 
of  the  different  index  numbers  themselves),  there  remain 
three  courses  to  pursue : 

(a)  to  employ  one  fixed  base  system,  using  Formula  353 
or  one  of  its  peers ; 

(6)  to  employ  the  chain  base  system,  using  Formula  353 
or  one  of  its  peers ; 

(c)  to  employ  the  broadened  base  system  (such  as 
Formula  6053). 

All  three  are  in  exceedingly  close  agreement. 

5.  Of  these  three  systems  the  chain  is  subject  to  cu- 
mulative error  and  ought  not  to  be  used  (unless,  possibly, 
as  supplementary  to  the  fixed  base  system). 

6.  Of  the  two  remaining  systems,  the  fixed  base  sys- 
tem (Formula  353)  is  somewhat  preferable  to  the  broad- 
ened base  system,  partly  because  it  is  slightly  closer  to 
the  best  blend  (7053)  and  partly  because  it  itself  is  not 
a  blend  at  all  and,  therefore,  not  blurred. 

7.  In  those  frequent  cases,  however,  where  the  data 
are  lacking  for  some  years  and  so  do  not  permit  of  using 
Formula  353,  or  its  rivals,  a  broadened  base  is  to  be 
used. 

8.  Two  broadened  base  formulae  are  practicable  for 


320         THE  MAKING  OF  INDEX  NUMBERS 

this  purpose;  the  aggregative  6053  and  the  geometric 
6023.  As  between  these  two,  while  both  are  good, 
Formula  6053  seems  clearly  the  better  because  there  is 
no  bias  even  if  only  two  years  are  included  in  the  base, 
or  even  only  one,  the  formula  then  reducing  to  53.  It 
often  happens  that  only  one  year's  quantities  are  known, 
in  which  case  Formula  23  or  53  must  be  used.  Formula 
23,  however,  is  not  usable  because  of  its  downward  bias, 
whereas  53  is  good,  practically  as  good  as  6053. 

§  12.  Historical 

The  fixed  base  system  has  always  been  the  principal 
method  of  presenting  index  numbers,  sometimes  the  first 
year  being  used  as  the  base  and  sometimes  a  series  of 
years.  The  broadened  base  system  has  been  in  common 
use  beginning,  apparently,  with  Soetbeer  and  Laspeyres. 
Professor  Alfred  Marshall  suggested  the  chain  system 
in  the  Contemporary  Review,  March,  1887,  and  in  the 
same  year  Professor  Edgeworth  and  the  Committee  on  this 
subject,  of  which  he  was  secretary,  recommended  the  chain 
system  to  the  British  Association  for  the  Advancement  of 
Science.  Walsh  advocated  and  adopted  it  in  his  book, 
The  Measurement  of  General  Exchange  Value.  Professor 
A.  W.  Flux  discussed  the  effect  of  changing  bases  in  a 
paper  in  the  Manchester  Literary  and  Philosophical 
Society,  1897,  and  ten  years  later,  in  the  Quarterly  Journal 
of  Economics,  discussed  the  chain  method,  but  without 
using  that  term.  The  term  "  chain  "  seems  first  to  have 
been  used  by  me  in  the  Purchasing  Power  of  Money,  in 
1911,  where  I  commended  it,  unduly,  as  I  now  believe.1 

1  Besides  the  historical  sections  scattered  through  the  book,  of  which  the 
above  is  the  last,  the  reader  will  find  in  Appendix  IV  a  brief  sketch  of 
"  Landmarks  in  the  History  of  Index  Numbers." 


CHAPTER  XV 
SPEED  OF  CALCULATION 

§  1.   Time  Studies 

HITHERTO  we  have  ignored  the  very  practical  question 
of  speed  and  ease  of  calculation.  Table  44  gives  the 
results  of  time  studies  for  calculating  the  index  numbers 
of  prices  by  the  various  formulae.  The  table  is  constructed 
on  the  assumption  of  36  prices  and  quantities1  supplied 
to  the  computer.  He  is  furnished  with  a  computing 
machine  and  logarithmic  tables.  The  time  required  to 
construct  index  numbers  for  either  prices  or  quantities  for 
the  years  1914-1918  by  Formula  51  (fixed  base)  is  taken 
as  unity.  In  the  case  of  the  particular  computer  who 
gave  himself  to  these  time  studies,  Formula  51  required 
56  minutes.  As  he  was  probably  slightly  more  rapid 
than  the  average  computer,  we  may  think  of  the  time  for 
51  as  one  hour,  and  of  all  the  other  figures  in  the  table  as, 
therefore,  representing  hours.  In  every  case  the  time  of 
calculation  was  that  required  to  calculate  the  five  index 
numbers,  to  two  decimal  places.2  The  absolute  times 
would  be  different,  of  course,  if  there  were  a  different 
number  of  commodities,  a  different  number  of  years,  or  a 
different  decimal  figure  to  be  calculated.  But  the  figures 
given  in  the  table  are  all  relative  to  the  time  of  calculating 
Formula  51  (or  151)  and  this  relative  time  would  not  be 

1  Except  in  the  case  of  the  simples,  for  which  no  quantities  are  needed, 
and  in  the  case  of  Formula  9051,  for  which  it  is  assumed  that  guessed  round 
weights  (1,  10,  100,  and  1000)  are  supplied. 

2  Except  for  the  modes  which  were  calculated  only  to  the  decimal 
point.    They  could  not  be  calculated  beyond  the  decimal  point  by  the 
rough  method  here  used. 

321 


322 


THE  MAKING  OF  INDEX  NUMBERS 


greatly  affected  by  any  changes  in  the  number  of  commod- 
ities, or  of  years,  or  of  decimal  points  to  be  computed. 

TABLE  44.     RANK  IN  SPEED  OF   COMPUTATION   OF 
FORMULA 


FORMULA  No. 

TIME  OF  COMPUTATION  AS  MULTIPLE  OF 
TIME  REQUIRED  BY  FORMULA  51  (FIXED 
BASE)  TAKEN  AS  UNITY 

RANK  IN  SPEED  OF 
COMPUTATION 

Fixed  Base 

Chain 

(Fixed  Base) 

5343 

64.3 

64.5 

109 

5307 

62.1 

62.2 

108 

5333 

51.5 

51.6 

107 

1303 

45.3 

45.5 

106 

345 

44.6 

44.6 

105 

5323 

44.2 

44.3 

104 

1343 

42.7 

42.8 

103 

4353 

39.4 

39.5 

102 

335 

38.1 

38.3 

101 

3353 

37.8 

*     37.9 

100 

7053 

37.5 

99 

343 

37.3 

37.5 

98 

245 

37.1 

37.3 

97 

247 

ii 

n 

M 

1333 

36.3 

36.4 

96 

307 

35.5 

35.6 

95 

1323 

35.1 

35.2 

94 

309 

34.8 

35.0 

93 

1353 

34.5 

34.7 

92 

235 

33.9 

34.0 

91 

237 

tt 

u 

u 

225 

31.9 

32.0 

90 

227 

« 

M 

n 

207 

31.7 

31.8 

89 

215 

« 

ii 

« 

126 

31.6 

31.7 

88 

325 

31.5 

31.6 

87 

323 

31.3 

31.4 

86 

333 

30.9 

31.0 

85 

146 

29.7 

29.8 

84 

108 

29.3 

29.4 

83 

243 

29.2 

29.0 

82 

1124 

29.1 

29.3 

81    • 

249 

28.4 

28.6 

80 

1123 

28.0 

28.1 

79 

1144 

27.9 

28.0 

78 

241=341 

27.1 

27.2 

77 

1143 

26.6 

26.8 

76 

SPEED  OF  CALCULATION 

TABLE  44  (Continued) 


323 


FORMULA.  No. 

TIME  OF  COMPUTATION  AS  MULTIPLE  OP 
TIME  REQUIRED  BY  FORMULA  51  (FIXED 
BASE)  TAKEN  AS  UNITY 

RANK  IN  SPEED  OF 
COMPUTATION 

Fixed  Base 

Chain 

(Fixed  Base) 

136 

26.5 

26.6 

75 

125 

26.1 

26.3 

74 

233 

26.0 

25.8 

73 

1104 

25.3 

25.5 

72 

239 

25.2 

25.4 

71 

1004 

24.9 

25.0 

70 

1014 

u 

ii 

« 

1134 

24.7 

24.8 

69 

145 

24.3 

24.4 

68 

1103 

24.2 

24.3 

67 

1154 

24.1 

24.3 

66 

107 

23.8 

24.0 

65 

1003 

23.7 

23.9 

64 

1013 

u 

u 

« 

229 

23.6 

23.8 

63 

1133 

23.4 

23.5 

62 

144 

23.0 

24.8 

61 

124 

23.0 

23.1 

60 

143 

22.8 

24.4 

59 

209 

22.8 

23.0 

58 

213 

(4 

14 

M 

123 

22.7 

22.9 

57 

3154 

u 

M 

fi 

26 

22.5 

22.6 

56 

28 

M 

M 

M 

223 

21.4 

23.5 

55 

46 

21.1 

21.3 

54 

48 

M 

« 

u 

135 

21.1 

21.2 

53 

301 

21.0 

21.2 

52 

4154 

20.8 

20.9 

51 

231=331 

20.6 

20.8 

50 

110 

20.5 

20.7 

49 

109 

20.4 

20.5 

48 

134 

19.8 

21.6 

47 

4153 

19.6 

19.8 

46 

133 

19.5 

21.2 

45 

36 

19.5 

19.7 

44 

38 

(( 

ii 

14 

1153 

18.7 

18.9 

43 

324         THE  MAKING  OF  INDEX  NUMBERS 
TABLE  44  (Continued) 


FORMULA  No. 

TIME  OF  COMPUTATION  AS  MULTIPLE  OF 
TIME  REQUIRED  BY  FORMULA  51  (FIXED 
BASE)  TAKEN  AS  UNITY 

RANK  IN  SPEED  OF 
COMPUTATION 

Fixed  Base 

Chain      . 

(Fixed  Base) 

8 

18.4 

18.6 

42 

16 

" 

ii 

14 

30 

18.2 

18.3 

41 

221=321 

17.6 

17.6 

40 

3153 

17.3 

17.4 

39 

25 

17.0 

17.9 

38 

27 

" 

" 

u 

29 

17.0 

17.2 

37 

44 

16.9 

17.0 

36 

50 

14 

" 

14 

6023     ('13-'16) 

16.5 

16.5 

35 

24 

16.1 

18.3 

34 

45 

15.7 

15.9 

33 

47 

" 

14 

14 

49 

" 

14 

14 

201 

" 

U 

U 

211 

14 

" 

" 

34 

15.3 

15.4 

32 

40 

" 

" 

14 

2353 

14.9 

15.1 

31 

6023     ('13-14) 

14.6 

14.6 

30 

6023  ('13&'18) 

" 

14 

14 

10 

14.3 

14.4 

29 

353* 

ii 

14 

" 

8054 

u 

" 

14 

35 

14.1 

14.3 

28 

37 

u 

u 

14 

39 

14 

" 

" 

8053 

II 

ii 

14 

2154 

14.0 

14.1 

27 

42  =  142 

13.9 

14.1 

26 

7 

13.0 

13.1 

25 

9 

" 

14 

14 

15 

" 

14 

" 

32  =  132 

12.9 

13.1 

24 

43 

12.6 

15.9 

23 

102 

12.6 

12.7 

22 

14 

12.0 

13.4 

21 

22  =  122 

11.9 

11.9 

20 

23 

11.6 

17.2 

19 

*  Identical  with  103,  104,  105,  106,  153,  154,  203,  205,  217,  219,  253,  259,  303,  305. 


SPEED  OF  CALCULATION 


325 


TABLE  44  (Continued) 


FORMULA.  No. 

TIME  OF  COMPUTATION  AS  MULTIPLE  OF 
TIME  REQUIRED  BY  FORMULA  51  (FIXED 
BASE)  TAKEN  AS  UNITY 

RANK  IN  SPEED  OF 
COMPUTATION 

Fixed  Base 

Chain 

(Fixed  Base) 

33 

11.0 

14.3 

18 

2 

10.5 

10.6 

17 

12 

u 

« 

it 

2153 

9.6 

9.8 

16 

54  =  4  =  5  = 

8.7 

8.9 

15 

18  =  19  =  59 

41  =  141 

8.5 

8.6 

14 

251=351 

7.8 

7.8 

13 

31  =  131 

7.5 

7.6 

12 

101 

7.4 

7.6 

11 

13 

6.6 

13.1 

10 

6053    ('13-'  18) 

6.5 

6.5 

9 

21  =  121 

6.4 

6.4 

8 

6053    ('13-'  16) 

6.1 

6.1 

7 

6053    ('13-'  14) 

5.6 

5.6 

6 

52  =  152 

5.5 

5.5 

5 

53=3  =  6  = 

5.3 

8.9 

4 

17=20  =  60 

1 

5.1 

5.3 

3 

11 

(4 

a 

M 

9051 

2.0 

2.0 

2 

51  =  151 

1.0 

1.0 

1 

§  2.   Comments  on  the  Table  of  Speed  of  Computation 
of  Formulae 

It  will  be  seen  that  the  first  prize  for  speed  goes  to 
Formula  51 ;  to  calculate  this  requires  only  one  hour. 
The  booby  prize  is  captured  by  a  mode,  5343 ;  this  re- 
quires 64.3  hours.  All  the  other  formulae  occupy  the  107 
intermediate  ranks. 

Our  ideal,  Formula  353,  requiring  14.3  hours,  ranks 
twenty-ninth.  In  speed  it  surpasses  all  the  twelve  other 
formulae  mentioned  in  Chapter  XI  as  rivaling  353  in 
accuracy.  One  of  the  13,  next  to  the  slowest  in  the  whole 


326         THE  MAKING  OF  INDEX  NUMBERS 

table,  is  Formula  5307,  requiring  62.1  hours.  Another, 
the  closest  competitor  with  353  for  the  place  of  honor 
for  accuracy,  Formula  5323  —  the  best  product  of  the 
geometric  type  —  requires  44.2  hours,  or  over  three  tunes 
as  long  as  353. 

Among  other  ranks  in  the  table  we  note,  beginning  near 
the  top,  or  slow,  end,  Formula  7053,  requiring  37.5  hours  ; 
1123,  one  of  Walsh's  favorites  for  accuracy,  28  hours; 
Lehr's4154,  20.8  hours;  1153,  another  favorite  of  Walsh, 
18.7  hours;  6023,  the  favorite  of  Professors  Day  and 
Persons,  16.5  hours  (when  four  years  are  combined  in 
the  broadened  base)  or  14.6  hours  (when  two  years  are 
combined).  All  these  take  longer  than  353  (14.3  hours). 

Among  those  requiring  less  time,  the  one  I  would 
especially  note  is  Formula  2153,  which  our  table  of  rank 
in  accuracy  shows  to  be  practically  identical  with  353. * 
The  tune  for  Formula  2153  is  only  9.6  hours.2  Formula 
6053  (with  a  four  years'  base)  requires  only  6.1  hours  (as 
against  16.5  for  its  rival,  6023).  Formula  53  requires 
only  5.3  hours,  and  9051,  when  only  round  weights,  multi- 
ples of  10,  are  used,  needs  but  2  hours. 

The  chain  system  usually  requires  five  or  ten  minutes' 

1  For  proof  see  Appendix  I  (Note  to  Chapter  XV,  §  2). 

8  Professor  Persons  ("Fisher's  Formula  for  Index  Numbers,"  Review 
of  Economic  Statistics,  May,  1921,  p.  104)  gives  some  time  tests  for  his 
Formula)  6023  and  353,  which  give  very  different  results  from  those  of  the 
tables  here  given.  There  are  two  reasons  for  this  difference.  In  the  first 
place,  Persons's  comparison  between  Formulae  6023  and  353  apparently 
omits  the  preliminary  work  of  calculating  the  weights  for  6023  and  so  does 
not  give  a  complete  comparison.  Our  figures  show  that  Formula  6023 
(four  year  base)  requires  16.5  hours  and  353,  14.3  hours  —  a  small  differ- 
ence, but  in  favor  of  353. 

The  second  point  is  that  Formula  2153  can  be  used  as  a  short  cut  for  353, 
reducing  the  time  to  9.6  hours,  or  nearly  half  of  that  for  6023,  for  which  no 
corresponding  short  cut  is  available. 

Why  Persons's  time  estimate  for  353  chain  should  be  double  that  of  353 
fixed  base,  I  do  not  understand.  In  any  time  study  I  have  made,  the  dif- 
ference between  these  two  is  much  smaller. 


SPEED  OF  CALCULATION  327 

more  time  than  the  fixed  base  system  although  in  a  few 
cases  it  actually  requires  less  (because  of  certain  items 
being  duplicated  in  that  system  and  so  needing  to  be 
calculated  but  once). 

It  will  be  noted  that,  in  many  cases,  the  most  accurate 
index  numbers  require  very  little  time  for  calculation 
while  the  least  accurate  require  a  great  deal  of  time. 
Thus  the  modes  are  very  time-consuming,  and  this  de- 
spite the  fact  that  they  are  worked  out  only  up  to  the 
decimal.  If  they  were  accurately  worked  out  by  formulae 
instead  of  roughly  calculated  by  ocular  inspection,  and 
if  they  were  carried  to  the  same  two  decimal  places  as 
are  used  for  the  other  formulae,  the  times  consumed  would 
be  several-fold  more  than  the  figures  entered  in  the  table. 
As  it  is,  the  slowest  formula  is  a  mode,  5343 ;  the  other 
modes  in  order  are :  Formulae  345,  1343,  343,  245,  247, 
146,  243,  249,  1144,  341,  1143,  145,  144,  143,  46,48,  all  in 
the  slower  half  of  the  list,  and  of  the  remaining  modes,  44, 
45,  47,  49,  142,  43,  141,  none  can  boast  of  speed,  —  even 
the  fastest  of  them  (141  or  41)  ranking  fourteenth.  Nor 
are  the  medians  as  fast  as  tradition  has  led  us  to  believe. 
The  modern  use  of  calculating  machines  has  put  the 
median  to  shame.  The  fastest  median,  the  simple  For- 
mula 31  (or  131),  stands  twelfth,  requiring  7J  hours. 

For  practical  use,  even  when  the  highest  accuracy  is 
demanded,  we  never  need  to  go  beyond  the  fastest  16 
formulae.  The  sixteenth  formula  is  2153,  which,  we  have 
seen,  is,  to  all  intents  and  purposes,  always  identical  with 
the  ideal  353.  And  of  these  first  sixteen  the  only  ones 
which  have  any  valid  claim  to  be  used  in  actual  practice 
are  Formula  2153  (sixteenth,  requiring  9.6  hours),  31 
(twelfth,  requiring  7.5  hours),  21  (eighth,  requiring  6.4 
hours),  6053  (seventh,  requiring  6.1  hours),  53  (fourth, 
requiring  5.3  hours),  and  9051  (second,  requiring  2  hours). 


328          THE  MAKING  OF  INDEX  NUMBERS 

It  will  be  noted  that  several  of  the  index  numbers 
used  or  recommended  by  others  are  not  included  in  the 
above  list.  The  simple  arithmetic  index  number,  Formula 
1,  stands  well  as  to  speed  of  calculation,  ranking  third 
and  requiring  only  5.1  hours.  But,  as  we  have  seen,  it 
ranks  among  the  "  worthless  "  formulae  in  accuracy.  If 
a  simple  index  number  is  really  necessary,  because  of  lack 
of  data  for  weighting,  Formula  21  and  31  are  far  more 
accurate  than  Formula  1  and  do  not  take  very  much 
longer  to  calculate  (6.4  and  7.5).  Usually,  however,  the 
round  weight  Formula  9051,  which  is  shorter  to  calculate 
and  at  the  same  time  more  accurate  than  Formula  1, 
can  be  used.  Formula  53,  which  is  still  more  accurate  and 
requires  but  a  trifle  more  time,  can  be  used  if  quantities 
are  known.  Formula  54  need  almost1  never  be  used. 
It  has  often  been  recommended,  but,  in  accuracy,  it  is 
exactly  as  far  from  the  ideal  353  on  one  side  as  53  is  on  the 
other,  while  53  can  be  calculated  nearly  twice  as  quickly 
as  54. 

Formula  6023,  recommended  by  Professors  Day  and 
Persons  of  Harvard,  is  inferior  both  in  accuracy  and 
speed  to  6053  and  2153.  Formulae  1123,  1153,  and 
1154,  formerly  recommended  as  the  theoretically  best  by 
Walsh,  are  probably  not  quite  as  accurate  as  2153  (as 

1  The  only  case  where  Formula  53  cannot  be  used  in  place  of  54  is  when 
the  base  weights  (go's)  are  lacking  while  the  current  year  weights  ((ft's)  are 
available.  The  only  instance  of  such  a  case  which  has  come  to  my  atten- 
tion is  that  of  foreign  exchange.  The  Federal  Reserve  Bulletin  now  pub- 
lishes an  index  number  of  the  Foreign  Exchanges  relatively  to  their  "pars." 
These  pars  (e.g.  $4.86f  for  sterling)  are  the  base  prices  (po's).  But  there 
are  no  corresponding  base  quantities  (QQ'S)  since  the  "  base,"  in  this  case, 
is  of  no  historical  year;  in  fact  for  some  countries  the  "par"  was 
never  historically  realized.  But  the  current  quantities  ((ft's)  are  avail- 
able. Here  Formula  54  is  indicated  (or  one  of  its  equals,  4,  5,  18,  19). 
There  is  scarcely  any  other  unbiased  formula  available.  At  present  the 
Federal  Reserve  Bulletin  uses  Formula  29,  which  has  an  upward  bias,  —  and 
a  large  one  when,  as  at  present,  the  exchanges  have  a  wide  dispersion. 


SPEED  OF  CALCULATION  329 

shown  in  our  table  of  ranks  and  in  our  discussion  in 
Chapter  XII),  and  require  from  twice  to  three  times  as 
long  to  calculate.  In  his  last  book  Walsh  has  also  rec- 
ommended1 Formula  2153  for  adoption  in  practice,  as 
well  as  353  as  probably  the  most  perfect  theoretically. 

The  most  important  result  of  this  chapter  is  that 
Formula  2153  may  be  used  as  a  short-cut  method  of  com- 
puting Formula  353,  it  being  so  close  an  approximation 
to  353  as  practically  to  be  identical.  It  gives  almost 
the  same  result  (within  less  than  one  part  in  2500)  and 
in  9.6  hours  instead  of  14.3  hours.  It  should,  therefore, 
in  practice  be  used  when  yearly  data  permit. 

When  yearly  data  are  incomplete,  we  should  use  one 
of  the  following  formulae :  6053,  53,  9051,  21,  31,  according 
to  the  completeness  of  available  data,  as  set  forth  in 
Chapter  XVII,  §  8. 

1  For  further  discussion,  see  Chapter  XVII,  §  8. 


CHAPTER  XVI 

OTHER  PRACTICAL  CONSIDERATIONS 

§  1.  Introduction 

WE  have  studied  the  accuracy  of  the  various  possible 
formulae  for  index  numbers  and  their  comparative  speeds 
of  computation.  These  are  the  two  chief  considerations 
in  constructing  an  index  number.  But  the  problem  of 
accuracy  was  not  fully  covered ;  for  our  study  was  confined 
to  the  question  of  the  formula  and  did  not  cover  the  data 
that  went  into  the  formula.  Hitherto,  in  this  book, 
by  the  lt  accuracy  "  of  an  index  number  has  been  meant 
its  accuracy  as  a  measure  of  the  average  movement  of 
the  given  set  of  prices  (or  quantities,  as  the  case  may  be) 
We  have  found,  for  instance,  that  Formula  353  enables  us 
to  measure  the  average  changes  of  the  prices  of  the  36 
specified  commodities  within  less  than  one  part  in  a  thou- 
sand. Yet  the  index  numbers  which  have  been  thus  com- 
puted and  found  to  possess  a  marvelously  high  degree  of 
accuracy,  as  a  measure  of  the  movements  of  those  com- 
modities, do  not,  of  course,  pretend  to  any  such  degree  of 
accuracy,  as  a  measure  of  the  movement  of  the  prices  of 
all  the  commodities,  perhaps  many  hundreds,  which  we 
would  wish  to  be  represented.  To  obtain  such  precision 
in  measuring  the  general  movement  of  all  the  prices  we 
would  need  to  have  and  to  use  them  all.  Practically,  such 
completeness  of  data  is  never  possible.  We  must  content 
ourselves  with  samples.  We  want  to  find,  therefore,  an 
index  number  constructed  from  a  relatively  small  num- 
ber of  commodities  which  shall  measure,  as  accurately 

330 


OTHER  PRACTICAL  CONSIDERATIONS        331 

as  possible,  the  movement  not  only  of  this  small  number 
included,  but  also  of  those  excluded. 

Thus  are  opened  up  two  new  lines  of  investigation 
with  regard  to  the  accuracy  of  index  numbers,  namely, 
the  influence  of  (1)  the  assortment  of  samples  and  (2)  the 
number  of  samples.  Each  of  these  subjects  offers  a  field 
of  study  which  has  scarcely  yet  been  touched.  I  shall 
try  here  merely  to  utilize  what  has  already  been  accom- 
plished by  Mitchell,  Kelley,  Persons,  and  others,  and  to 
urge  that  their  important  work  be  followed  up  either  by 
them  or  by  other  investigators. 

These  two  subjects  are  probably  quite  as  important 
as  the  choice  of  the  formula.  Certain  it  is  that,  when  the 
number  of  samples  used  is  small,  an  unwise  choice  can 
spoil  the  result.  It  is  also  doubtless  true  that  even  the 
best  available  assortment  and  number  of  commodities 
cannot  yield  the  same  degree  of  accuracy  as  the  merely 
mathematical  accuracy  of  the  formulae.  I  venture  to 
express  the  guess  that,  when  thoroughgoing  studies  are 
made  in  these  two  fields,  it  will  be  concluded  that  we  can 
seldom  reduce  the  errors,  or  fringe  of  uncertainty,  of  our 
index  numbers  to  less  than  one  or  two  per  cent.  As  com- 
pared with  such  errors,  small  though  they  be,  the  errors 
which  we  have  found  present  in  the  formulae  are  quite 
negligible.  In  short,  in  view  of  the  rather  rough  work 
required  of  it,, the  formula  (whether  it  be  353  or  any  other 
among  over  a  score  of  the  best  formulse)  may  be  regarded 
as  a  perfectly  accurate  instrument  of  measurement. 

§  2.  The  Assortment  of  Samples 

What  is  a  wise  assortment  depends  greatly  on  the 
purpose  of  the  index  number.  If,  for  instance,  the  pur- 
pose is  to  represent  the  general  movement  of  wholesale 
prices  of  foods  in  the  United  States,  there  should  be  more 


332         THE  MAKING  OF  INDEX  NUMBERS 

samples  of  meats  than  of  fish  and  more  of  cereals  than  of 
garden  vegetables.  The  assortment  should  also  include 
representatives  of  the  various  stages  of  production. 
Again,  if  all  stages  are  included  in  one  line  of  goods,  e.g. 
wheat,  flour,  and  bread,  the  corresponding  stages  should 
be  included  in  other  lines  such  as  corn,  hogs,  and  pork. 

The  price  movements  of  any  raw  material  and  its 
finished  products,  such  as  cotton  and  cotton  goods,  pig  iron 
and  wire  nails,  or  wheat  and  flour  will  tend  to  resemble 
each  other.  On  the  other  hand,  there  will  be  a  cross- 
wise correspondence  between  all  raw  materials  as  con- 
trasted with  all  finished  products  —  cotton,  pig  iron,  and 
wheat,  on  the  one  hand,  moving  somewhat  alike,  while 
cotton  goods,  wire  nails,  and  flour  will  move  somewhat 
alike.  As  shown  by  Mitchell,1  the  raw  materials  fluctuate 
more  widely  than  do  the  finished  products.  Again,  goods 
finished  for  consumers  for  family  use  have  a  resemblance 
to  each  other  as  compared  with  goods  finished  for  indus- 
trial use,  the  latter  fluctuating  more  than  the  former. 
Every  group  having  any  distinctive  character  should  be 
represented  in  due  proportion  to  the  others.  The  price 
quotations  should  also  be  fairly  assorted  geographically. 

This  process  of  fair  sampling  is  intimately  related  to 
the  process  of  fair  weighting,  for  which,  in  fact,  it  may 
roughly  be  used  as  a  substitute.  The  Canadian  Depart- 
ment of  Labor  and  the  British  Board  of  Trade  endeavor 
to  obviate  the  need  of  any  specific  weighting  by  represent- 
ing the  important  groups  of  goods  by  a  large  number 
of  commodities,  or  series  of  quotations,  while  representing 
the  unimportant  commodities  by  a  small  number  and 
then  taking  a  simple  average.  By  means  of  such  precau- 
tions, a  simple  index  number  virtually  loses  its  freakish 
weighting,  and  becomes  roughly  equivalent  to  a  weighted 

1  Bulletin  284,  United  States  Bureau  of  Labor  Statistics,  pp.  44,  45. 


OTHER  PRACTICAL  CONSIDERATIONS        333 

index  number.  The  simple  geometric  formula  (21)  is 
thereby  made  nearly  as  good  as  the  well-weighted  formula 
(1123),  a  vast  improvement,  and  Formula  1  brought  nearer 
to  1003,  an  improvement,  but  not  so  vast,  for  the  upward 
bias  remains,  though  the  freakishness  has  gone.  Thus,  the 
Canadian  index  number  has  the  bias  of  Formula  1003  while 
the  British  Board  of  Trade  index  number  has  very  nearly  the 
excellence  of  1123.  Bradstreet's  index  number  (Formula 
51),  thanks  to  a  good  selection  of  data,  has  also  been 
converted  from  what  would  otherwise  be  a  worthless  index 
number  into  a  fairly  good  index  number,  being  virtually 
9051,  or  a  close  approach  to  53.  Without  such  precautions 
great  distortion  occurs. 

During  the  Civil  War  the  Economist  index  number 
became  erratic  because,  out  of  22  commodities,  no  less  than 
four  were  cotton  and  cotton  products.  As  the  Civil  War 
raised  cotton  prices  enormously,  the  Economist  index 
number  showed  in  1864  a  rise  of  45  per  cent  over  the 
price  level  of  1860,  whereas  Sauerbeck's  index  number  of 
45  commodities  showed  for  the  same  period  only  a  12 
per  cent  rise  (both  series  being  recomputed  from  1860 
as  base) .  Again,  in  the  Aldrich  Report  of  1893,  the  simple 
average  included  25  kinds  of  pocket  knives,  making  pocket 
knives  25  times  as  important  as  wheat,  or  corn,  or  coal. 

At  best,  however,  such  multiplication  of  commodities 
is  only  a  rough  substitute  for  actual  weighting.  On  the 
other  hand,  even  when  weights  are  used  they  need  to  be 
adjusted  to  fit  in  with  the  numbers  of  commodities  included 
under  the  various  groups.  Thus  the  War  Industries  Board, 
having  included  seven  groups  (foods,  clothing,  rubber- 
paper-fiber,  metals,  fuels,  building  materials,  chemicals) 
subdivided  into  50  classes  (nearly  1500  separate  commodi- 
ties or  series  of  quotations),  proceeded  to  weight  them  in 
two  stages.  In  the  first  place,  each  commodity  was 


334         THE  MAKING  OF  INDEX  NUMBERS 

weighted  according  to  statistics  or  estimates  of  the  pro- 
duction or  volume  of  business  done  in  that  commodity. 
Then,  in  the  second  place,  inasmuch  as  some  of  the  50 
classes  were  more  fully  represented  than  others,  i.e. 
were  represented  by  a  larger  number  of  commodities, 
the  classes  which  were  meagerly  represented  in  number 
of  commodities  were  the  more  liberally  weighted  to  com- 
pensate. The  weights  first  assigned  to  them  as  individual 
commodities  were  magnified  or  multiplied  by  factors 
called  "  class  weights  "  to  make  them  represent  more 
adequately  the  large  class  to  which  they  belong.  This, 
or  some  equivalent  procedure,  should  always  be  employed 
where  the  highest  accuracy  is  desired. 

In  short,  either  insufficient  weights  should  be  compen- 
sated for  by  duplicating  samples  (as  in  the  Canadian  and 
Board  of  Trade  index  numbers),  or  insufficient  samples 
should  be  compensated  for  by  additional  weighting  (as 
in  the  War  Industries  Board  index  numbers).  Except 
as  a  substitute  for  weighting,  samples  need  not  be  multi- 
plied greatly.  In  fact,  where  it  is  desired  to  save  labor  by 
restricting  the  number  of  commodities,  those  selected 
should  be  so  assorted  as  to  differ  from  each  other  hi  charac- 
ter as  much  as  possible  rather  than  to  resemble  each  other 
as  much  as  possible.  As  Professor  Kelley  says,  the  prices 
included  should  be  correlated  not  so  much  with  each 
other  as  with  those  excluded.1  Where  the  samples  are 
thus  well  selected,  the  index  number  will  not  only  rep- 
resent well  the  price  movements  of  the  commodities  in- 
cluded, but  also  those  excluded,  usually  the  larger  group. 

§  3.  The  Basis  of  Classification 

As  Professor  Mitchell  points  out,  there  is  no  consistent 
basis  of  classification  in  the  grouping  employed  by  the 

l" Certain  Properties  of  Index  Numbers,"  Quarterly  Publication  of 
the  American  Statistical  Association,  pp.  826-41,  September,  1921. 


OTHER  PRACTICAL  CONSIDERATIONS        335 

United  States  Bureau  of  Labor  Statistics  and  others. 
Sometimes  the  basis  is  physical  appearance  (e.g.  as  in  the 
case  of  "  metals "),  use  served  (e.g.  "  house  furnishing 
goods "),  place  of  production  ("  farm  products "),  the 
industry  concerned  ("  automobile  supplies  ")j  etc.  Mitch- 
ell thinks,  on  the  whole,  that  the  most  useful  classifica- 
tions are  raw  versus  manufactured;  the  raw  being  sub- 
divided into  farm  crops  and  animal,  forest,  and  mineral 
products,  and  the  manufactured  being  subdivided  into 
goods  for  personal  consumption,  such  as  sugar,  and  goods 
for  business  consumption,  such  as  tin  plates. 

I  venture  to  express  the  opinion  that  we  shall  ulti- 
mately find  two  chief  bases  or  groups  for  classifying  goods. 

(1)  We  need  a  basis  for  setting  off  the  particular  field 
which  the  index  number  is  to  represent.     Since  this  may 
be  any  field  whatever  in  which  we  are  interested,  the  basis 
for  including  or  excluding  commodities  may  be  physical 
appearance,  use  served,  or  anything  else,  according  to  the 
field  to  be  studied.     For  instance,  a  leading  paper  manu- 
facturer has  constructed  for  use  in  his  business  an  index 
number  of  the  costs  involved  hi  the  manufacture  of 
paper.     These  comprise  wood  pulp,  labor,  and  all  other 
items  entering  into  that  cost. 

(2)  On  the  other  hand,  the  basis  on  which,  within 
the  particular  field  thus  marked  out,  the  samples  should 
be  assorted  is  none  of  those  bases  above  mentioned  but 
rather  the  behavior  of  the  prices.    All  behaviors  should  be 
fairly  represented.     In  the  paper  manufacturer's  index 
number  of  costs,  both  labor  and  wood  pulp  should  be 
represented,  not  because  they  are  so  widely  different  in 
physical  nature,  but  because  the  price  of  wood  pulp  and 
the  price  of  labor  behave  differently.    If  it  were  true 
that  they  always  rose  and  fell  together  a  sample  of  either 
would  serve  perfectly  for  both. 


336         THE  MAKING  OF  INDEX  NUMBERS 

One  of  the  most  interesting  kinds  of  index  numbers  is 
Professor  Persons' s  new  index  number  for  use  as  a  barom- 
eter of  trade.  In  this  case  the  selection  of  the  ten  com- 
modities included  is  based,  not  on  any  of  the  usual 
criteria,  but  on  their  previous  behavior  in  relation  to  the 
business  cycle. 

§  4.  The  Number  of  Quotations  Used 

Ideally,  the  quotations  should  be  as  inclusive  as  possible 
of  the  quotations  properly  belonging  to  the  class  being 
studied.  In  reality,  however,  we  are  restricted  by  expense 
or  other  practical  obstacles.  If  the  assortment  is  good,  y 
the  number  is  not  very  important.  The  War  Industries 
Board  used  1474  commodities,  or  series  of  quotations. 
But  the  resulting  index  number  differs  only  slightly 
(seldom  by  one  per  cent)  from  that  of  the  United  States 
Bureau  of  Labor  Statistics  for  about  300  commodities. 

Wesley  C.  Mitchell,  in  Bulletin  284  of  the  United  States 
Bureau  of  Labor  Statistics,  has  compared  the  index 
number  of  the  Bureau  for  about  250  commodities  with  the 
index  number  for  145,  50,  40,  and  25  commodities,  taking 
care  to  retain  a  similar  representation  of  the  various  con- 
stituent groups  of  commodities  in  the  cases  of  the  145 
and  50  commodities,  but  making  the  40  representative 
on  another  principle,  and  choosing  the  25  at  random. 
He  found  that  the  145  index  differed  on  the  average  from 
the  index  of  the  Bureau  of  Labor  Statistics,  by  less  than 
one  per  cent,  the  50  index  by  less  than  two  and  one  half  per 
cent,  the  40  index  by  less  than  5.4  per  cent,  and  the  25  index 
(taken  in  two  ways)  by  less  than  four  and  three  per  cent. 

I  have  made  a  similar  comparison  of  various  numbers 
of  commodities  from  the  list  published  weekly  in  Dun's 
Review.  Beginning  with  200  commodities  and  succes- 
sively halving,  I  have  taken  the  sub-lists  of  100  commodi- 


OTHER  PRACTICAL   CONSIDERATIONS        337 


ties,  50,  25,  12,  6,  and  3,  so  selected  as  to  be,  so  far  as 
possible,  equally  and  fairly  representative  of  the  various 
classes  of  commodities  in  exchange.1  These  were  calcu- 
lated (relatively  to  1913  as  a  base)  by  Formula  53  (or  3). 
The  results  are  plotted  in  Chart  61.  They  show  a  rather 
surprising  resemblance.  Taking  200  as  a  standard  of 
comparison,  and  gauging  the  closeness  of  the  others  to 
this  by  the  average2  of  their  deviations  from  it,  we  find  the 
following  figures : 

TABLE  45.     DEVIATIONS  FROM   200 
COMMODITIES   INDEX 


NUMBER  OF  COMMODITIES 
INCLUDED  IN  INDEX 

DEVIATIONS 
(PER  CENTS)' 

100 

1.78 

50 

2.05 

25 

1.61 

12 

2.64 

6 

4.31 

3 

3.65 

1  The  similarity  in  assortment  is,  of  course,  necessarily  rough.  It  is 
impossible,  for  instance,  to  assort  three  or  six  commodities  so  as  to 
include  a  sample  in  every  one  of  the  eight  classes  used  for  the  200 
commodities.  The  actual  assortments  are  shown  in  the  following  table : 

PERCENTAGES    OF   AGGREGATE    VALUE   OF    THE    200,    100,    50.    25,    12,    6, 
AND   3  COMMODITY   INDEXES   RESPECTIVELY,   IN  EACH   GROUP 


is 

| 

Si 

*  § 

|| 

S  6 

INQ  MA- 

ALS 

CALS  AND 

res 

m 

j 

i1 

I1 

1 

la 

f 

l|§ 
3s* 

11 

n 

3  ± 

5° 

Is 
1° 

3B 

200 

27.48 

20.80 

11.39 

8.61 

18.47 

6.54 

3.85 

2.86 

100. 

100 

27.18 

22.73 

9.29 

10.94 

18.22 

5.13 

3.35 

3.16 

100. 

50 

30.57 

29.19 

12.75 

3.45 

13.04 

6.85 

2.35 

1.80 

100. 

25 

23.69 

30.06 

13.72 

5.24 

16.64 

8.11 

1.09 

1.45 

100. 

12 

29.73 

34.76 

9.40 

6.90 

13.73 

3.35 

.76 

1.37 

100. 

6 

40.40 

35.97 

5.80 

0.00 

13.28 

4.55 

0.00 

0.00 

100. 

3 

55.75 

25.92 

0.00 

0.00 

18.33 

0.00 

0.00 

0.00 

100. 

2  Calculated  as  the  square  root  of  the  average  of  the  squares  of  the  deviations. 


338 


THE  MAKING  OF  INDEX  NUMBERS 


From  this  table  and  Chart  61,  it  is  clear  that  the  mere 
number  of  commodities  is  of  only  moderate  importance. 
A  small  number  may  be  nearly  as  good  as  a  large  number 
provided  they  be  equally  well  selected  or  assorted. 

According  to  the  theory  of  probabilities,  the  probable 
error  of  the  mean  of  any  number  of  observations  is  in- 

£ ffect  of 

Number  of  Commodities 
on  Index  Nos. 


22/^y 

£sSs 

25  COMMODITIES 


_/00  COMMODITIES 
200  COMMODITIES 


1921 
APR.  8 


1922 
JAN.I3 


J'MS  AU&5  NOV.II 

CHART  61.  Comparing  the  index  numbers  of  200,  100,  50,  25,  12,  6, 
and  3  commodities,  each  group  having  roughly  similar  proportions  of  farm 
products,  foods,  clothing,  fuel  and  lighting,  metals,  building  materials, 
drugs,  and  miscellaneous. 

versely  proportional  to  the  square  root  of  the  number. 
This  rule  would  apply  here  if  all  commodities  were  inde- 
pendent and  equally  important.  We  could  then  say, 
for  instance,  that  50  commodities  would  show  twice  the 
error  which  four  times  that  number,  or  200  commodities, 
would  show,  and  the  latter,  in  turn,  twice  that  of  800. 
By  this  law  of  the  square  root,  accuracy  increases  very 
slowly  with  an  increase  in  number. 


OTHER  PRACTICAL  CONSIDERATIONS        339 


In  actual  fact  the  improvement  in  accuracy  with  an 
increase  in  the  number  of  commodities  is  even  slower  than 
this  rule  would  lead  us  to  expect.  From  the  preceding 
table  of  deviations  I  think  it  may  be  inferred  by  rough 
averages1  that,  in  order  to  reduce  the  error  by  half  we 
must  multiply  the  number  of  commodities  not  by  four  but 
by  thirty-five.  If  this  be  true,  the  index  number  of  the 
War  Industries  Board  with  its  1366  commodities  is  only 
twice  as  accurate  as  an  index  number  formed  from  40 
commodities,  other  things  equal. 

This  slowness  of  improvement  in  index  numbers  with  an 
increase  in  the  number  of  commodities  is  largely  because 
the  number  of  commodities  does  not  represent  their 
importance  or  weights.  These  weights  for  the  100,  50, 
25,  12,  6,  and  3  commodity  groups  (in  dollars  and  in  per 
cents  of  the  weights  of  the  200)  are  as  follows : 

TABLE  46.  COMPARISON  OF  THE  AGGREGATE  VALUE  OF 
THE  100,  50,  25,  12,  6,  AND  3  COMMODITIES  WITH  THE 
AGGREGATE  VALUE  OF  THE  200  COMMODITIES 


No.  OF  COMMODITIES 

AGGREGATE  VALUE 
(IN  MILLIONS  OF  DOLLARS) 

AGGREGATE  VALUE 
(IN  PER  CENTS) 

200 

18266 

100 

100 

12079 

66 

50 

6572 

36 

25 

4331 

24 

12 

3284 

18 

6 

2416 

13 

3 

1751 

10 

If  we  use  these  weights  instead  of  the  number  of  the 
commodities  the  resulting  law  of  increasing  accuracy 
with  increase  in  weights  of  commodities  included  is  more 
nearly  in  accord  with  that  required  by  the  theory  of 
probability.  As  Table  46  shows,  when  a  small  number 

1  Obtained  by  plotting  the  table  of  standard  deviations  in  relation  to 
the  number  of  commodities  on  doubly  logarithmic,  or  ratio  chart,  paper. 


340         THE  MAKING  OF  INDEX  NUMBERS 

of  commodities  is  used,  we  naturally  choose  the  most 
important,  which  means  those  having  the  greatest  weights. 
If,  now,  we  calculate  the  relationship  between  the  errors 
of  the  index  numbers  of  100,  50,  25,  12,,  6,  and  3  commodi- 
ties, on  the  one  hand,  and,  on  the  other,  not  the  total 
number  of  commodities  but  their  total  weights,  we  find 
that,  on  the  average,  in  order  to  reduce  the  error  by  half, 
we  must  multiply  the  total  weight  of  the  commodities 
by  ten,  whereas  probability  theory  requires  four. 

Incidentally,  by  extending  graphically  these  rough 
laws  connecting  error  and  the  number  or  weight  of  com- 
modities, it  may  be  estimated  that  the  probable  error  of 
the  index  number  of  the  200  commodities  as  samples  as 
compared  with  an  index  number  calculated  from  an 
absolutely  complete  set  of  commodities  is  about  1£  per 
cent.1  But  in  order  to  obtain  a  trustworthy  empirical 
formula  we  would  need  very  much  fuller  data  than 
those  here  given.  I  hope  someone  will  make  a  thorough 
enough  study  of  this  subject  to  obtain  such  a  for- 
mula. 

An  index  number,  really  valuable,  has  been  computed 
for  as  few  as  10  commodities, — that  recently  constructed 
by  Professor  Persons  to  be  used  for  forecasting.  Seldom, 
however,  are  index  numbers  of  much  value  unless  they 
consist  of  more  than  20  commodities ;  and  50  (the  number 
of  classes  used  by  the  War  Industries  Board)  is  a  much 
better  number.  After  50,  the  improvement  obtained  from 
increasing  the  number  of  commodities  is  gradual  and  it  is 
doubtful  if  the  gain  from  increasing  the  number  beyond 
200  is  ordinarily  worth  the  extra  trouble  and  expense. 

1  This  same  result  obtains  whether  the  numbers  or  the  weights  are 
used.  For  another  (Kelley's)  method  of  reckoning  this  probable  error, 
see  Appendix  I  (Note  to  Chapter  XVI,  §4).  As  there  shown,  Kelley's 
method  yields,  as  its  result,  a  little  less  than  1  per  cent  in  one  case  and  1.3 
per  cent  in  another. 


OTHER  PRACTICAL  CONSIDERATIONS        341 


§  5.  Errors  in  the  Data 

It  is,  of  course,  vital  that  the  original  data  shall  be  as 
accurate  as  possible.  That  is,  the  markets  used,  the  sources 
of  quotations,  and  the  collecting  agency  should  be  the  most 
reliable  and  authoritative.  Nevertheless,  the  net  effect 
on  the  index  number  of  inaccuracies  in  the  original  data 
is  smaller  than  would  naturally  be  supposed,  especially 
if  a  large  number  of  commodities  are  used.  If  there  be 
100  commodities  and  an  average  or  typical  group  of  ten 
among  them  are  each  ten  per  cent  too  high,  the  net  effect  on 
the  index  number  is  to  make  it  only  one  per  cent  too  high! 
And  the  chances  against  all  ten  thus  erring  in  the  same 
direction  is  negligible.  The  errors  would  probably  largely 
offset  each  other,  so  that  the  probable  error  in  the  index 
number  which  would  result  from  ten  typical  commodities, 
out  of  100,  being  each  ten  per  cent  wrong,  but  some  too 
high  and  others  too  low  at  random,  would  not  be  one  per 
cent,  but  only  about  one  fourth  of  one  per  cent.  If 
every  one  of  the  100  commodities  is  subject  to  an  error 
of  ten  per  cent  in  either  direction  at  random,  the  net 
resultant  error  in  the  index  number  would  probably  not 
be  over  two  and  one  half  per  cent. 

From  such  surprising  examples  we  see :  (1)  that  even 
rough  data  are  valuable  if  we  have  enough  of  them,  and 
(2)  that,  under  conditions  of  ordinary  and  reasonable 
accuracy  of  the  data,  the  inaccuracies  which  actually 
enter  have  a  negligible  influence  on  the  result,  probably 
less  than  one  tenth  of  one  per  cent  in  the  case  of  such  an 
index  number  as  that  of  the  United  States  Bureau  of 
Labor  Statistics. 

What  has  been  said  applies  to  the  price  data  (for  an 
index  number  of  prices).  The  quantity  data,  which  are 
needed  only  for  the  weights,  require  even  less  accuracy. 


342         THE  MAKING  OF  INDEX  NUMBERS 

As  is  shown  in  Appendix  II,  §  7,  the  effect  of  a  change 
in  a  weight  is  only  a  small  fraction  of  that  of  a  change  in  the 
price  relative.  If  the  data  for  any  or  all  of  the  weights 
were  wrong  by  50  or  100  per  cent,  the  effect  on  the  index 
number  would  seldom  amount  to  one  per  cent. 

§  6.  The  Errors  of  Four  Standard  Index  Numbers 

We  have  now  seen  that  the  accuracy  of  an  index  number 
depends  upon  four  circumstances : 

(1)  the  choice  of  the  formula, 

(2)  the  assortment  of  items  included, 

(3)  the  number  of  items  included, 

(4)  the  procuring  of  the  original  data. 

At  present,  the  chief  source  of  error  in  standard  or  current 
index  numbers  is  in  the  formula.  This  book  shows  that 
this  source  of  error  can  (if  full  data  are  available)  be  elimi- 
nated entirely  —  or,  to  be  exact,  can  be  reduced  to  much 
less  than  a  tenth  of  one  per  cent. 

We  may  now  summarize  the  whole  subject  of  the 
degree  of  accuracy  of  index  numbers  by  citing  four  actual 
examples :  the  index  numbers  of  wholesale  prices  of  the 
War  Industries  Board,  the  United  States  Bureau  of 
Labor  Statistics,  the  Statist's  or  Sauerbeck's,  and  Pro- 
fessor Day's  index  numbers  of  prices  and  quantities  of 
12  crops.  In  each  case  I  shall  estimate  or  guess  at  the 
errors  due  to  each  of  the  four  sources  of  possible  error  and 
the  extent  to  which  such  errors  were  avoidable. 

The  War  Industries  Board  index  number,  which  is  for 
the  years  1913-1918,  is  probably  the  most  accurate  index 
number  ever  constructed  owing  to  the  huge  number  of 
commodities  included  and  the  fact  that  the  data  for 
quantities  are  available. 

1.  The  error  hi  this  index  number  due  to  errors  of  the 
formula  (53)  is  usually  less  than  one  fourth  of  one  per 


OTHER  PRACTICAL  CONSIDERATIONS        343 

cent,  but  reaches  about  one  half  of  one  per  cent  for  1918  * 
(the  figure  used  being  below  the  ideal,  353). 

2.  The  error  due  to  errors  in  the  assortment  of  items 
included   (corrected  by  class  weighting)   is,  I  imagine, 
always  less  than  one  per  cent. 

3.  The  error  due  to  the  number  of  commodities  (over 
1300)  not  being  complete  is,  I  imagine,  less  than  one  half 
of  one  per  cent. 

4.  The  error  due  to  errors  in  the  original  data  is  pre- 
sumably less  than  one  tenth  of  one  per  cent. 

The  net  error  due  to  all  four  sources  is,  I  imagine, 
usually,  if  not  always,  less  than  one  per  cent.  All  of  the 
errors  were  doubtless  unavoidable  excepting  that  due  to 
the  choice  of  the  formula,  and  this  probably  accounts 
for  perhaps  a  third,  or  a  half,  of  the  net  error.  That  is, 
this  most  precise  of  index  numbers  might  have  been  twice 
as  precise  as  it  is  had  Formula  353  (or  any  of  its  peers) 
been  used  as  the  formula  instead  of  53.  Had  this  been 
done  it  would  have  been  worth  while  to  use  a  figure  beyond 
the  decimal  point.  It  is  a  pity  that  the  highest  available 
degree  of  precision  was  not  reached,  as  such  a  good  oppor- 
tunity for  calculating  353  seldom  occurs,  owing  to  the  non- 
availability in  most  cases  of  statistics  of  yearly  quantities. 

We  may  next  consider  Day's  index  numbers  of  prices 
and  quantities  of  12  crops. 

1.  As  to  the  formula,  or  instrumental  error,  the  calcula- 
tions of  Professor  Persons  comparing  Professor  Day's 
(6023)  with  the  "  ideal  "  (353)  shows  an  error  hi  the  price 
index  of  usually  less  than  one  fourth  of  one  per  cent, 
exceeding  one  per  cent  only  once,  when  it  was  1.6  per  cent. 

1  As  judged  from  the  90  raw  materials  for  which  the  War  Industries 
Board  publishes  the  full  data  needed  for  calculating  Formula  353. 
Charts  for  Formulae  53  and  54  (the  latter  calculated  by  me)  for  these  90 
commodities  are  given  in  Chapter  XL  Formula  353,  of  course,  exactly 
splits  the  difference  between  53  and  54. 


*  344         THE  MAKING   OF  INDEX  NUMBERS 

For  quantities,  the  error  is  usually  less  than  one  per  cent, 
the  maximum  being  1.5  per  cent. 

2.  As  to  assortment,  I  can  only  guess  roughly  that  the 
error  from  this  source  would  be  inside  of  one  or  two  per 
cent. 

3.  As  to  the  number  of  commodities,  I  would  guess, 
say,  two  per  cent. 

4.  As  to  accuracy  of  data,  I  would  guess  that  the  index 
number  would  not  be  affected  more  than  one  per  cent. 

The  total  net  error  is  probably  usually  within  three  or 
four  per  cent,  although  if  all  the  errors  happened  to  be  in 
the  same  direction  and  all  large  they  might  make  a  total 
of  five  or  six  per  cent.  I  assume  that  all  of  these  errors 
are  unavoidable,  except  that  due  to  choosing  a  formula 
with  a  slight  downward  bias.  Had  Formula  353  been 
chosen  instead  of  6023,  the  error  would  have  been  reduced, 
but  seldom  by  as  much  as  one  per  cent.  While  the  gains 
in  accuracy  by  using  a  better  formula  would  be  small 
as  compared  with  the  errors  from  other  and  less  avoidable 
sources,  they  would  have  been  worth  while,  to  say  nothing 
of  the  gain  in  speed  of  computation.  As  indicated  in  Chap- 
ter XIV,  §  9,  the  formula  error  is  bound  in  the  future  to 
grow  indefinitely. 

Our  next  example  is  that  of  the  United  States  Bureau 
of  Labor  Statistics.  The  errors  are  probably  about  the 
same  as  for  the  War  Industries  Board : 

1.  Formula.     53  is  used,  erring,  say,  usually  less  than 
one  fourth  of  one  per  cent,  and,  at  most,  say,  one  half 
of  one  per  cent. 

2.  Assortment.     Say,  less  than  one  per  cent. 

3.  Numbers  of  commodities.    Say,  less  than  one  per 
cent. 

4.  Data.     Say,  less  than  one  tenth  of  one  per  cent. 
The  total  or  net  error  is  presumably  usually  within  one 


OTHER  PRACTICAL   CONSIDERATIONS        345 

or  two  per  cent,  almost  all  being  presumably  unavoidable 
—  that  from  formula  included,  owing  to  the  non-avail- 
ability of  yearly  quantities. 

Finally,  let  us  look  at  Sauerbeck's,  or  what  is  now  the 
Statist's,  index  number. 

1.  Formula.    This  error  is  of  two  parts,  that  due  to 
the  bias  of  the  arithmetic  type  and  that  due  to  the  freak- 
ishness  of  the  simple  weighting.     The  first  can  be  esti- 
mated with  considerable  certainty,  if  we  calculate  the 
standard  deviation  and  use  our  formula  connecting  the 
standard  deviation  and  the  bias.     I  have  worked  out  the 
standard  deviation  for  1920,  relatively  to  the  base  1867- 
1877,  for  the  45  commodities.     This  is  129  per  cent,  from 
which  we  know  that  the  upward  bias  is  36  per  cent.    For 
1913  the  standard  deviation  was  33  per  cent  and  the  bias 
4.1  per  cent  so  that,  not  only  is  the  Sauerbeck-Statist  in- 
dex number  for  1920  distorted  upwa-rd  by  this  cause  by  36 
per  cent  relatively  to  the  original  base,  but  it  is  distorted 
relatively  to  1913  by  31  per  cent.     As  to  the  error  of 
f  reakishness  of  weighting  this  may  be  said  to  be  practically 
the  same  thing  as  the  error  of  assortment. 

2.  Assortment.     Say,  one  per  cent. 

3.  Number  of  commodities.     Say,  one  or  two  per  cent. 

4.  Data.     Say,  one  tenth  of  one  per  cent. 

The  net  error  is  probably,  say,  35  to  40  per  cent.  In 
this  case  the  source  of  almost  all  the  error  is  the  bias  in 
the  formula  which  reaches  so  high  a  figure  partly  because 
of  the  long  lapse  of  time  since  the  base  period  and  partly 
because  of  the  great  dispersion  due  to  the  confusion  pro- 
duced by  the  World  War.  This  source  of  error  is,  of 
course,  avoidable.  This  Sauerbeck-Statist  index  number 
has  done  pioneer  work  and  deserves  that  respect  always 
due  to  long  and  faithful  service.  But  it  is  now  both  too 
old  and  too  old-fashioned  to  be  of  great  service  in  the  future. 


346         THE  MAKING  OF  INDEX  NUMBERS 

§  7.  A  New  Index  Number 

I  have  worked  out  a  new  index  number  of  wholesale 
prices  of  200  commodities  by  a  method  which  combines 
speed  of  computation  with  as  much  accuracy  as  the  data 
afford.  This  I  hope  later  to  publish  weekly.  The  data 
include  only  base  year  quantities  and  the  formula  used 
is  a  combination  of  53,  employing  base  year  quantities, 
and  9051.  For  the  28  most  important  commodities  the 
method  of  53  is  used,  i.e.  each  price  quotation  is  multiplied 
by  the  best  obtainable  statistical  figure  for  the  quantity 
marketed  of  that  commodity,  while  for  the  other  172 
commodities  the  round  figures  1, 10, 100,  or  1000  are  used, 
whichever  in  any  given  case  is  nearest  the  statistical 
figure.  No  sacrifice  of  accuracy  is  made  by  using  such 
round  figures  for  so  many  unimportant  commodities, 
as  I  have  proved  by  certain  tests.1 

In  this  way  we  avoid  the  necessity  of  having  labo- 
riously to  calculate  to  any  greater  degree  of  precision  than 
that  which  is  attainable.  This  saving  of  useless  labor,  is 
enormous.  To  calculate  this  index  number  of  200  com- 
modities, once  the  data  are  given,  requires  (for  the 
calculation  of  a  single  index  number)  only  two  and  a  half 
hours  as  contrasted  with  the  eight  hours  which  would  be 
required  if  all  of  the  200  statistical  weights  were  used.  As 
to  precision  reached,  I  believe  the  error  is  nearly  as 
small  as  that  of  the  United  States  Bureau  of  Labor 
Statistics,  the  total  error  being,  say,  usually  less  than 
two  per  cent. 

§  8.  An  Index  Number  should  be  Easily  Understood 

It  is  practically  important  that  an  index  number, 
besides  being  accurate  and  quickly  calculated,  should  be 

1  See  Appendix  I  (Note  to  Chapter  XVI,  §  7). 


OTHER  PRACTICAL  CONSIDERATIONS       347 

easily  understood.  In  this  respect  the  aggregatives 
have  an  obvious  advantage  over  all  other  types.  Any- 
one can  understand  Formula  53,  especially  if  the  base 
number  be  taken  not  as  100,  but  as  the  sum  of  values  ( Sp0<?o) 
in  the  base  year.  In  this  case  the  index  number  is  simply 
the  number  of  dollars  which  a  given  bill  of  goods  costs 
from  time  to  time.  Formula  54  is  almost  as  simple, 
being  merely  calculated  the  other  way  around.  Formula 
2153  is  next  in  simplicity,  the  bill  of  goods  being  the 
average  of  the  above  two.  Formula  353  is  a  little  harder 
for  the  man  in  the  street  to  understand,  but  is  intelligible 
as  the  mean  of  53  and  54.  These  are  easier  to  under- 
stand than  any  arithmetic  average,  still  easier  than  any 
harmonic,  and  far  easier  than  the  geometries. 

Another  advantage  of  the  aggregatives  is  that  of  sim- 
plicity and  ease  of  manipulation.  When  we  wish  index 
numbers  of  foods,  clothing,  .etc.,  as  subheads  under  a  gen- 
eral group  from  which  we  also  want  an  index  number,  the 
aggregative  is  the  most  easily  and  most  intelligibly  added, 
combined,  averaged,  and  otherwise  manipulated  —  far 
more  easily  than  the  medians,  in  particular. 

§  9.  Ranking  of  Formulae  by  Four  Criteria 

We  may  here  summarize  the  whole  subject  of  ranking 
the  formulae  in  Table  47,  in  the  third  column  of  which  I 
have  arbitrarily  ranked  the  20  simplest^  the  order  of  their 
simplicity  of  formula, — in  other  words,  their  intelligibility. 
The  three  other  columns  give,  likewise,  the  20  best  rank- 
ing formulae  in  respect  of  accuracy,  speed,  and  conform- 
ity to  the  so-called  circular  test.1 

1  The  order  of  "accuracy"  is  the  revised  order  given  in  Chapter  XII,  §  7. 
The  order  of  conformity  to  the  circular  test  here  given,  above  the  line  divid- 
ing those  which  fully  conform  from  the  rest,  is  also  revised  arbitrarily.  The 
order  given  in  Chapter  XIII,  §  9,  is  obviously  largely  accidental,  being  based 
on  only  four  data  for  each  formula. 


348 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  47.     (INVERSE)   ORDER  OF  RANK  OF  FORMULAE 

Of  the  20  First  in  Accuracy 
"    "     "       "    "Speed 
"    "     "       "     "Simplicity 
"    "     "       "    "  Circular  Test  Conformity 


ACCURACY 

SPEED 

SIMPLICITY 

CIRCULAR  TEST 
CONFORMITY 

8353 

22 

52 

2153 

1124 

23 

6023 

323 

1123 

S3 

23 

325 

124 

2\ 

353 

8054 

126 

12} 

8054 

8053 

123 

2153 

8053 

1323 

125 

54 

1153 

1353 

1154 

41 

9021* 

5323 

1153 

351 

9011* 

2353 

2154 

31 

9001* 

353 

2153 

101 

21 

21 

323 

13 

11 

22 

325 

6053 

2153 

51 

8054 

21 

6053 

52 

8053 

52 

54 

321 

1323 

53 

53 

351 

1353 

1} 

31 

6023 

5323 

111 

1 

6053 

2353 

9051 

9051 

9021* 

353 

51 

51 

9051  , 

*  Index  numbers  by  these  "rough  weight"  formulae  were  not  computed  for  this  book. 
Consequently  they  enter  into  the  competition  only  in  column  3  (and  one  of  them,  9021,  in 
column  4)  and  not  elsewhere,  where  computation  is  involved.  The  reason  for  omitting  their 
computation  is  the  impossibility  of  selecting  the  rough  weights.  Or  rather  there  are  an 
infinite  number  of  sets  of  rough  or  guessed  weights  which  might  be  used.  The  rough  weights 
in  the  case  of  9051,  on  the  other  hand,  are  definitely  ascertainable,  being  1,  10,  100,  etc. ; 
for  the  unique  idea  of  Formula  9051  is  a  minimum  of  calculation,  —  the  rough  and  ready 
summation  of  the  original  data  after  simply  shifting  decimal  points. 

Approximate  ranks  can  be  assigned  to  these  formula,  however.  With  any  reasonable 
selection  of  rough  weights,  Formulas  9001  and  9011,  if  computed  and  ranked  for  columns 
1,  2,  4,  would  be  too  inaccurate  to  find  a  place  among  the  20  most  accurate  (column  1), 
and  would  find  no  place  among  the  20  in  column  4,  but  would  take  rank,  in  speed,  below  the 
middle  of  column  2.  As  to  Formula  9021,  this  is  already  ranked  in  the  last  two  columns ; 
it  would  find  no  place  in  the  accuracy  list  (column  1)  but  would  find  a  place  in  the  upper 
half  of  the  speed  list  (column  2). 

We  see  that,  for  accuracy,  353  takes  first  rank,  for  speed 
in  computation,  51,  for  the  convenience  of  conformity  to 
the  so-called  circular  test,  any  of  the  ten  below  the  divid- 


OTHER  PRACTICAL  CONSIDERATIONS        349 

ing  line  in  the  fourth  column,  and,  for  simplicity,  51.  In 
this  table,  the  formulae  which  occur  but  once  are  itali- 
cized. As  none  of  these  are  near  the  goal  in  any  list 
they  certainly  need  never  be  used.  Eight  occur  three 
times  (21,  51,  353,  2153,  6053,  8053,  8054,  9051)  and  only 
one  (2153)  occurs  in  all  four  columns.  Taking  into 
account  accuracy,  speed,  ease  of  manipulation,  and 
intelligibility,  Formula  2153  seems,  on  the  whole,  to  take 
the  highest  rank  for  ordinary  practical  use. 

§  10.   Conclusions 

The  "  instrumental  error/7  or  error  in  the  index  number 
as  an  instrument  of  measure,  can  be  reduced  by  the  right 
choice  of  formula  so  low  as  to  be  negligible  as  compared 
with  the  errors  from  other  sources  —  particularly  the 
assortment  of  the  commodities  included  and  their  number. 
The  greater  the  number  of  commodities,  other  things 
equal  (including  assortment), the  more  accurate  it  is;  but 
the  increase  in  accuracy  is  very  slow,  requiring  perhaps  a 
thirty-five-fold  increase  in  numbers  to  cut  the  error  in 
two. 

Of  the  four  chief  sources  of  error,  formula,  assortment, 
number  of  commodities,  and  original  data,  the  two  first 
are  usually  most  at  fault.  The  error  in  the  Sauerbeck- 
Statist  index  number  today  reaches  over  35  per 
cent  from  the  first  source  alone.  If  an  index  number 
be  constructed  in  the  best  possible  way,  not  only  from 
accurate  data  and  with  an  adequate  number  of  commodi- 
ties, say,  several  hundred,  but  from  data  carefully 
assorted  for  the  purpose  in  view,  and  with  a  first-class 
formula,  such  as  353  or  2153,  it  can  probably  be  made 
accurate  within  close  to  one  per  cent. 

General  conclusions  as  to  ranking  are  stated  at  the  top 
of  this  page. 


CHAPTER  XVII 
SUMMARY  AND  OUTLOOK 

§  1.  Introduction 

AN  index  number  of  prices  is  intended  to  measure  such 
magnitudes  as  the  "  price  level  "  of  one  date  or  place 
relatively  to  that  of  another.  It  is  an  average  of  "  price 
relatives."  These  price  relatives  (or  movements  of  the 
prices  of  individual  commodities)  usually  disperse  or  scat- 
ter widely.  The  dispersion  or  scattering  of  the  price 
relatives  used  in  this  book  (for  the  years  1913  to  1918), 
was  especially  great.  Thus  the  price  of  wool  in  1918 
(relative  to  1913)  was  282  per  cent  and  that  of  rubber,  68 
per  cent.  Evidently  their  average  or  index  number 
(reckoned  arithmetically)  was  175  per  cent. 

Since  an  index  number  for  any  date  is  always  relative 
to  some  other  date  it  necessarily  implies  two  dates  or  peri- 
ods and  only  two.  When  we  calculate  a  series  of  index 
numbers  for  a  series  of  years  each  individual  index  number 
connects  one  of  the  years  with  some  other  year.  The 
usual  way  is  to  take  some  one  year,  such  as  the  first  year, 
as  the  "  base"  and  calculate  the  index  number  of  each 
other  year  relatively  to  that  common  base.  This  is 
called  the  "  fixed  base  system."  Another  way  is  that  of 
the  "  chain  "  system  by  which  the  index  number  of  each 
year  is  first  calculated  as  a  "  link  "  relatively  to  the  pre- 
ceding year  and  then  multiplied  by  all  the  preceding  links 
back  to  the  base  year. 

350 


SUMMARY  AND  OUTLOOK  351 

§  2.  Varieties  of  Types,  Weightings,  and  Tests 

There  are  only  six  types  of  index  number  formulae 
which  need  to  be  considered :  the  arithmetic,  harmonic, 
geometric,  median,  mode,  and  aggregative  —  all  defined  in 
Chapter  II.  Of  these,  the  mode  and,  in  general,  the  me- 
dian, may  be  ignored  as  too  sluggish  or  unresponsive  to 
small  influences  to  make  them  sensitive  and  accurate 
barometers  of  price  movements. 

As  shown  in  Chapters  III  and  VIII,  there  are  six  chief 
ways  of  "weigh ting  "  the  price  relatives  entering  into  any 
index  number  (except  the  aggregative)  viz.,  (1)  simple 
or  even  weighting,  each  price  relative  (like  the  282  and 
68)  being  counted  once ;  (2)  by  base  year  values  (desig- 
nated in  the  book  as  weighting  7),  wool  being  counted 
twice  to  rubber  once  if  the  value  of  the  wool  sold  in  the 
base  year  is  twice  that  of  the  rubber;  (3)  by  given 
year  values  (weighting  IV) ;  (4)  and  (5)  by  "  hybrid " 
values,  each  weight  being  formed  by  multiplying  the  price 
of  either  year  by  the  quantity  of  the  other  year  (weight- 
ings II  and  777) ;  and,  finally  (6)  by  crosses  or  means 
between  the  weightings  7  and  77,  or  77  and  777. 

Of  these  six  systems  of  weighting  the  middle  four  are 
fundamental  enough  to  tabulate : 

7  by  base  year  values  (pQqQ,  etc.) 

77  "  hybrid  «      (Poqly     «  ) 

777  "  hybrid  "     (Plq<>,     "  ) 

IV  "  given  year  "     (piqlf     "  ) 

In  the  case  of  the  aggregative  index  numbers,  since  the 
weights  (of  an  index  number  of  prices)  are,  in  this  case, 
merely  quantities  (not,  as  in  other  cases,  values),  we  have 
only  four  methods  of  weighting,  viz.,  (1)  simple ;  (2)  by 
base  year  quantities  (weighting  7) ;  (3)  by  given  year 


352         THE  MAKING  OF  INDEX  NUMBERS 

quantities  (weighting  I  V)  ;   and  (4)  by  a  cross  or  mean 
between  the  last  two  (/  and  IV). 

There  are  two  chief  tests  of  reversibility  for  an  index 
number  formula  (P)  :  First,  it  should  give  consistent 
results  if  applied  forward  (P0i)  and  backward  (PJO) 
between  the  two  dates  "  0  "  and  "  1  "  (i.e.  P01  X  PIO 
should  =  1).  This  has  been  called  Test  1,  or  the  time  re- 
versal test.  To  illustrate,  if  the  index  number  shows 
1918's  prices  average  twice  those  of  1913,  the  same  for- 
mula should,  when  applied  the  other  way  round,  show 
1913's  prices  to  average  half  those  of  1918.  Secondly, 
the  formula  should  give  consistent  results  if  applied  to 
prices  and  to  quantities  (i.e.  P0i  X  Qoi  should  =  V01, 


i.e.          l).     This  has  been  called  Test  2,  or  the  factor 


reversal  test.  To  illustrate,  if  we  know  that  the  total  value 
of  the  commodities  has  doubled  and  our  index  number 
of  prices  shows  that  prices  have,  on  the  average,  doubled, 
the  same  index  number  formula  should,  when  applied  to 
quantities,  show  that  quantities,  on  the  average,  have  re- 
mained the  same. 

In  short,  we  can  check  up  the  forward  and  backward 
index  numbers  by  the  principle  that  their  product  should 
be  unity,  and  we  can  check  up  the  price  and  quantity 
index  numbers  by  the  principle  that  their  product  should 
be  the  value  ratio. 

§3.  Bias 

But  many  kinds  of  index  numbers  do  not  thus  check 
up.  For  instance,  the  arithmetic  index  number  does  not. 
The  product  of  any  arithmetic  index  number,  taken  for- 
ward, multiplied  by  the  arithmetic  with  the  same  weights 
but  taken  backward,  fails  to  meet  Test  1  by  always  and 
necessarily  exceeding  unity  (i.e.  P01  X  PIO>  1)- 


SUMMARY  AND  OUTLOOK  353 

Thus,  if  we  designate  by  100  per  cent  the  1913  price  of 
each  of  the  36  commodities  used  in  this  book,  the  prices 
of  bacon,  barley,  beef,  etc.,  in  1917  are  respectively  193 
per  cent,  211  per  cent,  129  per  cent,  etc.,  the  simple  arith- 
metic average  of  which  figures  —  i.e.  the  simple  arith- 
metic index  number  for  1917  —  is  176  per  cent ;  while  if, 
reversely,  we  call  every  price  in  1917  100  per  cent,  the 
prices  of  bacon,  barley,  beef,  etc.,  in  1913  are  respectively 
52  per  cent,  47  per  cent,  77  per  cent,  etc.,  the  simple  arith- 
metic average  of  which  figures  is  63  per  cent.  But  these 
two  arithmetic  index  numbers,  176  per  cent  and  63  per 
cent,  are  mutually  inconsistent  since  the  ratio  of  176  to 
100  is  not  the  same  as  the  ratio  of  100  to  63,  i.e.  1.76  and 
.63  are  not  reciprocals.  In  other  words,  1.76  X  .63  is  not 
1.00,  but  is  1.11,  so  that  the  1.76  and  the  .63  are  too  big 
by  a  "  joint  error  "  of  11  per  cent,  or  about  5.5  per  cent 
apiece.  The  11  per  cent  is  their  "  joint  error  "  and  the 
5.5  per  cent  imputed  to  each  index  number  is  its  "up- 
ward bias/'  a  tendency  to  exaggerate  inherent  in  the 
arithmetic  process  of  averaging. 

Similarly,  the  harmonic  process  of  calculating  index 
numbers  has  a  downward  bias  (i.e.  P0i  X  PIO<  1). 

It  is  of  interest  to  observe  that  the  11  per  cent  or  other 
figure  calculated  for  the  "  joint  error  "  of  any  two  forward 
and  backward  index  numbers,  or  of  any  two  price  and 
quantity  index  numbers,  is  always  an  absolutely  true 
figure.  We  can  always  know  to  a  certainty  how  greatly 
the  product  of  the  two  index  numbers  errs.  But  —  and 
this  is  of  still  greater  interest  —  the  ascription  of  half  of 
the  joint  error  to  each  of  the  two  is  merely  a  guess,  based 
on  considerations  of  probability.  We  can  never  say  with 
certainty  how  far  wrong  any  one  index  number  may  be. 
The  "  absolutely  correct  "  figure  always  eludes  us.  We 
have  no  absolute  criterion  of  correctness  but  only  of  in- 


354         THE  MAKING  OF  INDEX  NUMBERS 

correctness.  Nevertheless  —  and  this  is  of  the  greatest 
interest  —  we  can,  on  grounds  of  probability,  narrow 
down  the  fringe  of  doubt  until  it  is  practically  negligible. 

Besides  the  above  mentioned  cases  of  bias  lurking  in 
two  types  of  index  numbers  —  the  arithmetic  and  the 
harmonic  —  there  is  another  sort  of  bias  pertaining  to 
certain  systems  of  weighting.  It  might  seem,  at  first  sight, 
that  any  of  the  six  systems  of  weighting  would  be  as 
likely  to  afford  errors  in  one  direction  as  the  other  —  that, 
for  instance,  the  use  of  base  year  values  as  weights  would 
be  no  more  likely  to  yield  a  small  index  number  than 
would  the  use  of  given  year  values,  nor  the  use  of  given 
year  values  to  yield  a  large  index  number  any  more  than 
that  of  base  year  values.  But  such  equal  liability  to  err 
in  either  direction  is  not  found.  Of  the  six  systems  of 
weighting  (applicable  to  all  the  types  of  index  numbers, 
except  the  aggregative),  only  the  simple  weighting  and 
the  cross  weighting  are  not  definitely  biased  in  some  one 
direction. 

As  to  the  other  four  weightings  (I,  77,  777,  77),  it 
was  shown  in  Chapter  V  that  the  formulae  with  weight- 
ings 7  and  777  have  necessarily  a  positive  joint  error,  as 
likewise  do  the  formulae  with  weightings  77  and  IV.  It 
was  also  shown  that  weightings  7  and  77  give  almost  iden- 
tical results,  as  also  do  777  and  IV.  Practically,  there- 
fore, the  four  systems  yield  only  two  results :  7,  77  and 
777,  IV  with  a  positive  joint  error  between  these  two.  If 
we  apportion  this  joint  error  equally,  we  may  say  that  7 
and  77  have  a  definite  downward  bias,  and  777  and  IV 
a  definite  upward  bias.  It  was  shown  that  the  reason 
these  weight  biases  exist  is  because  7  and  77  give  too 
much  weight  to  the  smaller  price  relatives  while  777  and 
IV  give  too  much  weight  to  the  larger  price  relatives. 

Bias  must  be  eliminated  in  order  to  obtain  a  good  index 


SUMMARY  AND  OUTLOOK  355 

number.  To  be  completely  free  of  bias  a  formula  of  un- 
biased type,  such  as  the  geometric,  needs  also  to  have  un- 
biased weighting,  such  as  cross  weighting.  A  biased  type, 
however,  can  be  remedied  by  the  use  of  an  oppositely 
biased  weighting,  or  vice  versa.  Thus,  in  the  case  of  an 
arithmetic  formula  weighted  by  base  year  values,  the 
upward  type  bias  is  offset  by  the  downward  weight  bias. 
Reversely,  in  the  case  of  a  harmonic  weighted  by  given 
year  values,  the  downward  type  bias  is  offset  by  the  up- 
ward weight  bias. 

Some  formulae,  however,  have  both  type  bias  and 
weight  bias.  Thus  the  arithmetic  formula,  if  weighted 
by  given  year  values,  has  a  double  dose  of  upward  bias 
(i.e.  both  the  upward  type  bias  inherent  in  the  arithmetic 
process  of  averaging  and  the  upward  weight  bias  inherent 
in  the  given  year  weighting).  Reversely,  the  harmonic 
formula,  if  weighted  by  base  year  values,  has  a  double 
dose  of  downward  bias  (i.e.  both  the  downward  type  bias 
inherent  in  the  harmonic  process  of  averaging,  and  the 
downward  weight  bias  inherent  hi  base  year  weighting). 

Other  formulae,  of  course,  have  just  a  single  dose  of  bias 
due  either  to  the  type  or  the  weighting.  Thus  the  geo- 
metric formula  weighted  by  given  year  values  has  simply 
the  upward  weight  bias  from  the  given  year  weighting 
without  any  type  bias,  while,  reversely,  the  geometric 
weighted  by  base  year  values  has  only  the  downward 
weight  bias  from  the  base  year  weighting  without  any 
type  bias.  Again  the  cross  weight  arithmetic  has  simply 
upward  type  bias  pertaining  to  the  arithmetic  process 
without  any  weight  bias,  while,  reversely,  the  cross  weight 
harmonic  has  only  the  downward  type  bias  of  the  harmonic 
without  any  weight  bias. 

The  bias  of  any  index  number  (whether  type  bias  or 
weight  bias,  or  both)  increases  with  the  dispersion  of  the 


356          THE  MAKING  OF  INDEX  NUMBERS 

price  relatives  and  in  a  rapidly  increasing  ratio.  Con- 
sequently a  biased  formula,  while  it  has  only  a  slight  error 
when  there  is  little  dispersion,  has  an  enormous  error  when 
(as  happens  with  the  lapse  of  tune)  there  is  a  great  dis- 
persion. 

As  to  the  aggregative,  the  two  types  of  weighting,  I 
and  IV,  are  not  biased.  The  aggregative  I  (Formula 
53 l  in  our  series  of  numbers)  is  known  as  Laspeyres' 
formula  and  the  aggregative  IV  (Formula  54)  is  known  as 
Paasche's  formula.  These  two  formulae  are  identical  re- 
spectively with  arithmetic  7  (Formula  3)  and  harmonic 
IV  (Formula  19),  as  well  as  with  certain  others. 

Although  only  two  (arithmetic  and  harmonic)  of  the 
six  types  of  formulae  and  only  four  (7,  77,  777,  IV)  of 
the  six  kinds  of  weighting  are  "  biased,"  i.e.  liable  to  err 
in  a  given  direction,  they  are  all  subject  to  some  error  and 
so  may  be  called  more  or  less  "  erratic."  When  a  formula 
is  especially  erratic  it  is  called  "  freakish."  The  mode 
and,  less  markedly,  the  median  are  freakish  types  and 
simple  weighting  is  freakish  weighting.  The  weighted 
aggregatives  are  only  slightly  erratic ;  the  joint  error  of 
the  forward  and  backward  aggregative  index  numbers  is 
very  small. 

§  4.  Derivation  of  Antithetical  Formulae 

By  means  of  the  two  reversibility  tests  we  find  that 
each  formula  has  its  special  "  time  antithesis  "  and  its 
special  "  factor  antithesis."  As  shown  in  Chapter  IV, 
the  time  antithesis  is  derived  by  reversing  the  timeSj  i.e. 
taking  the  index  number  backward,  and,  then  inverting 
the  result  (dividing  it  into  unity),  while  the  factor  antithe- 
sis is  derived  by  reversing  the  factors,  i.e.  taking  the  index 

1  For  the  mnemonic  system  of  numbering  the  various  formulae  see 
Appendix  V,  §  2. 


SUMMARY  AND  OUTLOOK  357 

number  for  quantities  instead  of  for  prices  and  then  di- 
viding the  result  into  the  value  ratio.    That  is,  the  time 

antithesis  of  any  index  number  formula,  P0i  is   ^-    while 

'Tio 

the  factor  antithesis  of  P0i  is        . 


By  these  processes,  applied  to  the  various  types  and  sys- 
tems of  weighting  already  described,  we  are  provided 
with  46  primary  formulae.  As  shown  hi  Chapter  VII, 
these  are  arrangeable  in  sets  of  four  each,  or  "  quartets  " 
(some  of  which  may  be  reduced  to  "  duets  ").  In  each 
quartet,  each  horizontal  pair  of  formulae  are  antitheses 
by  Test  1,  and  each  vertical  pan:  are  antitheses  by  Test 
2,  thus  forming  two  pau-s  of  time  antitheses  and  two 
pairs  of  factor  antitheses. 

These  46  primary  formulae  comprise  :  the  simples,  the 
weighted  7,  77,  777,  IV,  just  cited,  and  the  factor  antith- 
eses of  all  these.  Of  these  46  not  a  single  one  conforms 
to  the  factor  reversal  test,  and  only  four  (the  simple 
geometric,  median,  mode,  and  aggregative)  to  the  time 
reversal  test. 

§  5.  Rectification 

By  crossing  (i.e.  taking  the  geometric  mean  of)  any 
pair  of  time  antitheses,  we  derive  a  formula  which  conforms 
to  the  time  reversal  test;  and  by  crossing  any  pair  of 
factor  antitheses,  we  derive  a  formula  which  conforms  to 
the  factor  reversal  test  ;  while  by  doing  both,  we  derive 
a  formula  which  conforms  to  both  tests. 

Instead  of  crossing  formulae,  we  may,  as  already  stated, 
cross  then:  weights.  By  this  alternative  process  we  may 
also  derive  formulae  that  conform  to  Test  1.  The  two 
alternative  processes  do,  however,  present  certain  contrasts. 
For  instance,  hi  order  to  secure  conformity  to  Test  1, 
formula  crossing  must  be  accomplished  through  the  geo- 


358         THE  MAKING  OF  INDEX  NUMBERS 

metric  mean  (except  that,  in  two  cases,  those  of  the  geomet- 
ric and  aggregative  index  numbers,  the  aggregative  mean 
is  also  an  available  method).  Weight  crossing,  on  the 
other  hand,  may  be  accomplished  through  the  arithmetic, 
harmonic,  or  geometric  means  and,  of  these  three  methods, 
the  arithmetic  probably  yields  the  most  accurate  result. 

Any  given  cross  weight  formula  and  the  corresponding 
cross  formula  always  agree  very  nearly.  By  means  of 
crossing  formulse  we  increase  our  list  of  46  "  primary  " 
formulae  to  the  "  main  series  "  of  96  and,  by  crossing  the 
weights,  we  enlarge  the  series  to  134. 

By  rectification  any  bad  formula  can  be  reformed. 
Bias  can  be  elimrnated,  while  freakishness  can  be  reduced 
but  not  entirely  eHminated. 

§6.  Base  Shifting 

The  so-called  circular  test  requires  that,  in  a  given  series 
of  years  no  matter  which  year  is  taken  for  base,  the  re- 
sulting index  numbers  shall  stand  in  the  same  ratios  each 
to  each;  consequently,  if  we  calculate  index  numbers 
from  year  to  year,  or  from  place  to  place  around  a  speci- 
fied circuit  of  years  or  places,  we  shall  end  at  the  same 
figure  from  which  we  started.  But  this  circular  test  is, 
strictly,  not  a  fab-  test ;  for  shif ting  the  base  ought  to  change 
these  relations.  A  direct  comparison  between  two  par- 
ticular years  is  the  only  true  comparison  for  those  two 
years.  Comparisons  between  those  two  particular  years, 
via  other  years,  ought  not  necessarily  to  give  the  same  re- 
sult ;  on  the  contrary,  there  ought,  in  general,  to  be  a 
discrepancy  or  gap.  Nevertheless,  in  the  cases  of  our 
most  exact  formulse,  this  gap  is,  in  actual  fact,  found  to 
be  negligible,  being  only  a  small  fraction  of  one  per  cent. 
That  is,  the  circular  test,  although  theoretically  wrong, 
is  practically  fulfilled  by  the  best  formulse. 


SUMMARY  AND  OUTLOOK  359 

Consequently,  it  is  not  necessary,  in  practice,  to  calcu- 
late an  index  number  between  every  possible  pair  of  years. 
A  single  series  will  be  sufficiently  accurate  for  all  these 
inter-year  comparisons.  For  this  purpose,  we  may  use  the 
chain  system,  the  fixed  base  system,  the  base  being  one 
year  only,  or  a  broadened  fixed  base  system,  i.e.  one  hi 
which  the  base  is  an  average  over  several  years.  Of 
these  three,  the  chain  system  is  strictly  correct  only  for 
consecutive  years;  for  longer  comparisons  (i.e.  when 
reckoned  back  relatively  to  the  original  base),  it  is  subject 
to  cumulative  error.  Of  the  remaining  two,  the  broad- 
ened fixed  base  system  seems,  on  the  whole,  better  than 
the  fixed  base  system  with  its  single  year  as  base,  although 
we  may  often  be  forced  to  use  the  latter  for  lack  of  data 
necessary  for  calculating  any  broader  base.  Moreover, 
in  the  case  of  the  aggregative,  the  preference  is  inappre- 
ciable. 

Index  numbers  are  more  frequently  used  to  compare 
each  year  with  the  base  than  to  compare  successive  years. 
The  fixed  base  system,  when  used  for  comparing  two  years 
neither  of  which  is  the  base,  is  always  subject  to  some 
error.  But  this  error  is  usually  slight  and  is  not  cumula- 
tive. Only  for  long  or  for  very  dispersive  periods,  if  at 
all,  is  any  other  index  number  needed  in  addition  to  those 
of  the  fixed  base  system. 

§7.  Formulae  Compared 

To  find  the  best  formula  we  first  eliminate  as  "  freakish  " 
the  simples  and  their  derivatives,  and  the  modes  and  me- 
dians and  their  derivatives.  All  the  remaining  formulae 
fall  into  five  groups,  which  may  be  plotted  as  a  five-tined 
fork,  the  middle  tine  portraying  the  formulae  without 
bias,  the  two  tines  nearest  portraying  the  formulae  having 
a  single  dose  of  bias,  and  the  two  outer  tines  portraying 


360         THE  MAKING  OF  INDEX  NUMBERS 

the  formulae  having  a  double  dose  of  bias.  Eliminating 
all  biased  formulae,  we  have  remaining  only  those  on  the 
middle  tine,  47  in  number,  all  of  which  agree  closely  with 
each  other.  These  consist  of  rectified  formulae  and  of  the 
Formulae  53  and  54,  Laspeyres'  and  Paasche's.  Of  these 
47,  the  13  which  fulfill  both  tests  agree  with  one  another 
still  better.  Of  these  13,  the  "ideal"  Formula  353, 


J 
S 


^°  v     ^l>  is  at  least  equal  in  accuracy  and  is 


probably  slightly  superior  in  accuracy  to  any  of  the  others. 

This  Formula  ,353  is  demonstrably  correct  within  less 
than  one  eighth  of  one  per  cent  and  probably  within  a  hun- 
dredth of  one  per  cent,  as  a  measure  of  the  average  change 
of  the  given  data  (prices  or  quantities,  etc.)  between  the 
two  years  for  which  it  is  calculated.  In  other  words,  in 
the  case  of  Formula  353,  we  have  no  perceptible  "  instru- 
mental error  "  to  deal  with.  So  far  as  the  mere  question 
of  formula  is  concerned,  the  index  number  method  is  cer- 
tainly henceforth  to  be  recognized  as  possessing  as  high  a 
degree  of  precision  as  the  majority  of  physical  measures 
in  practical  use. 

But  there  is  no  thought  of  maintaining  that  353  is  the 
"  one  and  only  "  formula.  On  the  contrary,  a  chief  con- 
clusion is  that  all  index  numbers  which  are  not  freakish  or 
biased  practically  agree  with  each  other.  Even  the  freakish 
medians,  and  probably  also  the  more  freakish  modes, 
agree  with  the  good  ones  fairly  well  when  very  large  num- 
bers of  commodities  are  used.  In  all  others,  viz.,  the 
arithmetic-harmonic,  the  geometric,  and  the  aggregative, 
agreement  is  found  to  a  startling  degree.  In  other  words, 
the  idea  that  index  numbers  of  different  types  or  systems 
of  weighting  disagree  is,  in  general,  true  only  before  they 
are  "  rectified."  Those,  like  Pierson,  whose  studies  have 
led  them  to  distrust  and  abandon  index  numbers  as  worth- 


SUMMARY  AND  OUTLOOK  361 

less  have  simply  not  pushed  their  studies  far  enough. 
Nevertheless,  a  small  grain  of  truth  remains  in  Pierson's 
contention.  There  is  no  index  number  which  can  be  spoken 
of  as  absolutely  "  correct."  There  must,  theoretically, 
always  remain  a  fringe  of  doubt.  All  that  we  can  say 
with  certainty  is  that  this  fringe  of  doubt  instead  of  be- 
ing very  large,  as  Pierson  thought,  is,  for  the  "ideal" 
formula,  very  small  —  ordinarily  less  than  a  tenth  or 
even  a  hundredth  of  one  per  cent. 

§  8.  The  Eight  Most  Practical  Formulae 
We  have  seen  that  Formula  353  is  the  best  when  the 


utmost  accuracy  is  desired.    Formula  2153,  Z^°  +  qi\  Pl, 

2  wo  +  ft)  Po 

however,  which  will  seldom  appreciably  differ  in  its  re- 
sults from  353,  is  more  quickly  calculated. 

In  case  the  full  data  are  not  available  for  calculating 
Formula  353  or  2153,  but  data  are  available  for  calculat- 
ing 6053,  53,  or  54,  any  of  these  three  will  serve  excellently 
as  a  substitute  for  353.  If  data  even  for  Formula  53 
are  unavailable,  round-weights  may  be  guessed  at,  i.e. 
9051  may  be  used  as  a  makeshift  for  Formula  53. 

If  no  data  at  all  are  available  for  judging  the  relative 
weights  so  that  recourse  must  be  had  to  simple  formulae, 
the  simple  median  (Formula  31)  and  the  simple  geometric 
(Formula  21)  are  the  best,  with  possibly  a  slight  prefer- 
ence for  the  former  hi  most  cases.  The  simple  arithmetic 
(Formula  1)  should  not  be  used  under  any  circumstances, 
being  always  biased  and  usually  freakish  as  well.  Nor 
should  the  simple  aggregative  (Formula  51)  ever  be  used  ; 
in  fact  this  is  even  less  reliable. 

The  relative  accuracy  of.  these  eight  formulae  may 
roughly  be  given  as  follows  :  353  is  usually  correct  within 
one  hundredth  of  one  per  cent  ;  2153  is  usually  correct 


362         THE  MAKING  OF  INDEX  NUMBERS 

within  one  fourth  of  one  per  cent ;  6053, 53,  and  54  are  usu- 
ally correct  within  one  per  cent ;  9051  is  usually  correct 
within  three  per  cent;  21  and  31  are  usually  correct 
within  six  per  cent. 

These  eight  important  formulae  are  the  only  ones 
which  ever  need  to  be  used,  although  not  by  any  means  the 
only  ones  which  may  be  used.  Their  computation  and 
that  of  8053  are  exemplified  in  figures  in  Appendix  VI,  §  2. 

§  9.   Suggested  Application  to  the  United  States 

These  eight  formulae  are  to  be  used  according  to  the 
adequacy  of  the  data.  For  the  general  index  number  of 
the  United  States  Bureau  of  Labor  Statistics,  full  data  for 
quantities,  being  dependent  on  the  census  reports,  are 
available  only  once  in  ten  years.  Consequently,  Formula 
353  can  be  used  only  once  in  ten  years.  In  the  intervening 
period,  Formula  53  should  be  used  as,  in  fact,  it  is.  At 
the  close  of  each  decade  the  figure  reached  by  Formula  53 
can  be  checked  up  by  means  of  353  applied  to  the  new 
data  then  available.  The  discrepancy  may  then  be  pro- 
rated over  the  preceding  ten  years  and  these  corrected 
figures  be  substituted  in  all  future  publications  for  the 
figures  originally  obtained  by  Formula  53,  just  as  is  done 
with  population  figures. 

The  figures  for  Formula  53  should  be  calculated  by  the 
fixed  base  method,  as  at  present,  and  not  the  chain  sys- 
tem, so  that  the  discrepancy  at  the  end  of  the  decade  may 
be  a  minimum.  The  Formula  353  figures,  on  the  other 
hand,  being  calculated  between  successive  censuses, 
would  form  a  chain  system,  each  link  being  a  decade, 
although,  to  satisfy  scientific  curiosity,  it  would  be  well,  as 
each  new  census  appears,  to  calculate  from  each  new  census 
directly  to  all  the  preceding  censuses.  The  discrepancies 
which  would  be  found  would  inevitably  be  negligible. 


SUMMARY  AND  OUTLOOK  363 

§  10.  Critique  of  Formulae  Proposed  by  Others 
It  has  been  necessary  to  compare  many  varieties  of 
formulae  only  to  find,  in  the  end,  little  practical  use  for 
most  of  them.  Until  complete  comparisons  were  made  we 
could  not  be  sure  which  agreed  or  disagreed,  which  were 
correct,  or  which  lent  themselves  to  rapid  calculation. 

Of  the  25  formulae  mentioned  by  previous  writers  as 
possibly  valuable,  we  have  seen  that  the  following  ought 
never  to  be  used  because  of  bias:  1,  2,  9,  11,  23.  The 
following  ought  never  to  be  used  because  of  freakishness : 
41,  51,  52.  All  the  rest  may  be  used  under  various  cir- 
cumstances (as  to  availability  of  data)  as  may  also 
about  35  other  formulae  presented  in  this  book  for  the 
first  time.  All  these  usable  formulae  will  agree  under 
like  circumstances  with  the  seven  formulae  actually  rec- 
ommended as  the  most  practical. 

The  only  formulae  much  in  use  of  the  25  formulae  men- 
tioned by  previous  writers  are :  1,  21,  31, 51, 53, 6023,  6053, 
9021.  Of  these  eight,  Formula  21  or  Formula  9021,  now 
used  by  the  British  Board  of  Trade,  53  or  6053,  used  by 
the  United  States  Bureau  of  Labor  Statistics  and  the  Aus- 
tralian Bureau  of  Census  and  Statistics,  and  Formula 
23  or  6023,  used  by  Professors  Day  and  Persons  in  the 
Review  of  Economic  Statistics,  published  by  the  Harvard 
Committee  on  Economic  Research,  are  all  good,  although 
the  last  named  will  deteriorate  as,  with  the  lapse  of  time, 
the  base  period  is  left  very  far  away.  Of  the  other  five, 
the  most  thoroughly  objectionable  are  1  and  51,  although 
1  is  the  formula  most  often  used.  There  are  two 
objections  to  Formula  1,  the  simple  arithmetic,  viz. :  (1) 
that  it  is  "  simple, "  and  (2)  that  it  is  arithmetic !  —  that 
it  is  at  once  freakish  and  biased.  In  the  case  of  Sauer- 
beck's index  number,  for  instance,  the  bias  alone  reaches 
36  per  cent ! 


364         THE  MAKING  OF  INDEX  NUMBERS 

The  conclusions  of  the  present  book  depart  from  pre- 
vious thought  and  practice  in  fundamental  method. 
Hitherto  writers  have  been  debating  the  "  best  type  " 
(whether  arithmetic,  geometric,  or  median)  by  itself,  the 
"  best  weighting  "  by  itself,  and  the  bearing  on  these  of 
the  distribution  of  price  relatives.  But  from  our  study 
it  should  be  clear  that  it  makes  little  difference  what  type 
we  start  with,  or  what  the  weighting  is  (so  long  as  it  is 
systematic),  or  what  the  distribution  of  price  relatives 
may  be  so  long  as  we  "  rectify  "  the  formulae  and  so  elim- 
inate all  these  sources  of  distortion  or  onesidedness. 

Moreover,  even  if  we  do  not  thus  rectify  the  primary 
formulae  but  merely  choose  from  among  them,  our  study 
helps  us  do  the  choosing,  so  as  to  avoid  bias  and  minimize 
error.  Thus,  as  to  the  long  controversy  over  the  relative 
merits  of  the  arithmetic  and  the  geometric  types,  our  study 
shows  us  that  the  simple  geometric,  21,  is  better  than  the 
simple  arithmetic,  1,  but  that,  curiously  enough,  the 
weighted  arithmetic,  3,  is  better  than  the  weighted  geo- 
metric, 23. 

§  11.  Speed  of  Computation 

The  chief  practical  restriction  on  the  use  of  the  many 
fairly  good  formulae  is  imposed  by  the  time  required  to 
calculate  them.  No  formula,  for  instance,  surpasses 
appreciably  in  accuracy  Formula  5323  and,  were  it  as 
easily  calculated  as  its  equivalent,  353,  I  would  seriously 
suggest  5323  for  practical  use.  But,  on  a  test  problem, 
it  requires  44.2  hours  to  calculate  5323  while  Formula 
353,  which  yields  precisely  as  good  a  result,  requires  only 
14.3  hours,  and  2153,  which  yields  almost  as  good  a  re- 
sult, requires  only  9.6  hours. 

Besides  accuracy  and  speed  we  need,  in  practice,  to 
consider  two  other  qualities,  viz.,  conformity  to  the 


SUMMARY  AND  OUTLOOK  365 

so-called  circular  test,  and  simplicity,  or  intelligibility  to 
the  uninitiated.  The  best  practical  all-around  formula, 
taking  all  four  points  into  account,  —  accuracy,  speed, 
minimum  legitimate  circular  discrepancy,  simplicity  — 
is  the  Edge  worth-Marshall  formula,  2153. 

Formula  353  is  "  best  "  only  in  the  sense  of  accuracy, 
as  the  telescopes  in  the  great  observatories  are  best. 
But  smaller,  cheaper  telescopes,  spy  glasses,  and  opera 
glasses  still  have  their  uses.  No  one  would  want  a  Lick 
telescope  on  the  porch  of  his  summer  residence  or  at  the 
theater. 

§  12,  Two  Consequences  of  the  Agreement 
of  Index  Numbers 

Among  the  consequences  of  the  surprising  agreement 
between  the  various  legitimate  methods  of  calculating 
index  numbers  are  two  which  need  emphasis  here.  The 
first  is  that  all  discussion  of  "  different  formula  appropriate 
for  different  purposes  "  falls  to  the  ground.  The  second 
is  that,  the  supposed  differences  among  formulae  once 
banished,  the  real  problem  of  accuracy  is  shifted  to  the 
other  features  of  an  index  number,  —  the  assortment  of 
the  commodities  included,  their  number,  and  data. 

Errors  due  to  mere  insufficiency  of  number  are  relatively 
small,  while  those  due  to  inaccuracies  of  data  are  usually 
negligible,  even  though  trreselfriaccuracies  individually 
be  great.  Thus  the  figures  for  weights  in  particular  may 
usually  be  tenfold  or  one  tenth  of  the  true  figures  with- 
out appreciably  disturbing  the  accuracy  of  the  resulting 
index  number.  Henceforth,  the  effort  to  improve  the 
~  accuracy  of  index  numbers  must  center  chiefly  on  the 
assortment  of  the  items  to  be  included.  This  will  differ 
for  the  different  purposes  to  which  the  proposed  index 
number  is  to  be  put. 


366         THE  MAKING  OF  INDEX  NUMBERS 

§  13.  Current  Ideas 

How  do  the  conclusions  reached  in  this  book  differ 
from  previous  views  on  index  numbers  ?  Largely,  of  course , 
these  views  are  confirmed  and  supported  by  new  data. 
The  main  results  of  C.  M.  Walsh's  thoroughgoing  studies 
are  supported.  His  three  favorite  formulae,  advocated  in 
his  first  and  larger  book,  the  Measurement  of  General  Ex- 
change Value,  are  1123,  1153,  and  1154,  all  of  which  are 
"  superlative  "  in  our  hierarchy  of  index  numbers,  i.e. 
practically  peers  of  353.  He  also  advocated  (as  Marshall 
and  Edgeworth  did  before  him  and  as  I  do)  Formula  2153. 
In  his  second  book,  The  Problem  of  Estimation,  as  already 
indicated,  he  reached  independently  the  conclusion  that 
Formula  353  is  probably  the  king  of  all  index  number 
formulae.  In  like  manner,  the  conclusions  of  this  book 
support  and  are  supported  by  most  of  the  work  of  Jevons, 
Marshall,  Edgeworth,  Pigou,  Flux,  Knibbs,  Mitchell, 
Meeker,  Young,  Persons,  and  Macaulay. 

Yet  many  of  the  conclusions  are  new  and  of  these  sev- 
eral run  athwart  current  ideas.  The  concept  of  bias,  as 
it  applies  to  the  arithmetic  and  harmonic  types,  has  been 
implicitly  recognized  (though  not  specially  named)  by 
Walsh,  and,  to  some  extent,  by  others;  but,  as  applied 
to  systems  of  weighting,  it  is  new. 

One  of  the  points  which,  though  by  implication  recog- 
nized by  Walsh,  will  appear  as  new  to  almost  everyone 
else,  is  that  the  kind  of  weighting  befitting  any  index  num- 
ber is  different  for  different  types. 

Test  1  has  been  more  or  less  definitely  recognized,  but 
Test  2  is  new  and  no  index  number  hitherto  in  actual  use 
conforms  to  Test  2. 

Rectification  is  a  new  idea,  except  as  to  one  special 
case  (namely  rectification  relatively  to  Test  1  accomplished 


SUMMARY  AND  OUTLOOK  367 

by  means  of  weight  crossing).  Consequently,  many  of 
the  formulae  derived  in  the  processes  of  rectification  are 
new  and  several  of  these  new  formulae  are,  so  far  as  ac- 
curacy is  concerned,  practically  as  good  as  any  formula 
previously  suggested. 

The  conclusion  that  the  circular  test  is  theoretically 
wrong  is  entirely  new ;  that  it  is  nevertheless  practically 
right,  as  applied  to  all  good  index  numbers,  is  almost 
new ;  that  all  index  numbers  conforming  to  rational  stand- 
ards of  excellence  agree  to  a  nicety  is  new ;  that  the  par- 
ticular type  of  formulae  and  the  particular  weighting  of 
formulae  prior  to  rectification  and  the  particular  sort  of 
dispersion  or  distribution  of  the  relatives  to  be  averaged, 
are  unimportant,  and  that  only  the  criteria  of  goodness 
are  vitally  important,  is  new ;  finally,  that  in  selecting  an 
index  number  formula  the  purpose  to  which  it  is  put  is 
immaterial  is  practically  new. 

In  view  of  these  divergences  from  current  thought,  it 
is  not  surprising  that  the  conclusions  reached  often  col- 
lide with  current  practice. 

§  14.  The  Future  Uses  of  Index  Numbers 

If  the  conclusions  reached  are  correct,  some  of  the  meth- 
ods of  calculating  index  numbers  now  most  in  vogue 
should  be  discontinued.  It  is  high  tune  that  index  num- 
bers should  be  so  calculated  as  to  enable  us  to  get  out  of 
them  all  there  is  hi  them.  Their  use  is  rapidly  growing 
and  often  with  little  heed  paid  to  the  methods  of  making 
them.  When  they  are  made  rightly,  as  a  matter  of  or- 
dinary routine,  their  usefulness  will  be  greatly  increased 
and  may  be  extended  to  many  fields  scarcely  touched 
upon  as  yet. 

Thirty  years  ago  only  wholesale  price  indexes  were 
used  and  even  these  were  not  as  numerous,  as  widely 


368         THE  MAKING  OF  INDEX  NUMBERS 

known,  or  as  widely  used  as  today,  when  so  many  official 
agencies  and  so  many  trade  journals  publish  them.  In- 
dex numbers  of  retail  prices,  of  wages,  and  of  the  prices 
or  sales  of  stocks  were  rarities,  if  not  curiosities.  Today 
these  are  common.  In  Great  Britain  alone,  three  million 
laborers  have  their  wages  regulated  annually  by  an  index 
number  of  retail  prices.  We  have  numerous  index  num- 
bers of  the  stock  market,  even  in  daily  papers.  We  now 
have  also  index  numbers  of  the  cost  of  living,  of  the  mini- 
mum of  subsistence,  and  of  wages  in  terms  of  that  mini- 
mum. Good  index  numbers  of  the  quantities  of  goods 
produced,  consumed,  or  exchanged  are  also  comparatively 
new.  Beginning  with  the  crude  efforts  of  Rawson-Rawson 
a  generation  ago,  Kemmerer  hi  1907,  and  myself  in  1911, 
such  index  numbers  have  in  the  past  few  years  come  to 
have  considerable  statistical  value,  and  are  even  becoming 
differentiated  into  indexes  of  production,  manufacture, 
crops,  national  income,  imports,  exports,  barometers  of 
trade,  etc.  Another  recent  application  of  index  num- 
bers, now  current  in  at  least  five  countries,  is  that  of 
measuring  the  trend  of  the  foreign  exchanges. 

One  of  the  most  interesting  recent  developments  is  the 
application  of  index  numbers  to  special  industries,  such  as 
lumber  or  building  (e.g.  the  Aberthaw  Index  of  the  cost  of 
a  cement  building) ;  or  even  to  special  individual  busi- 
nesses, such  as  the  American  Writing  Paper  Company 
(e.g.  for  paper  production  costs) ;  or  even  to  special  de- 
partments in  an  individual  business  (e.g.  the  price  of 
textbooks  of  Henry  Holt  and  Company).  When  the 
business  statistician  begins  to  realize  the  usefulness  of 
this  device  in  his  own  business,  index  numbers  will  be 
found  sprouting,  right  and  left,  to  serve  the  purposes  of 
trade  journals,  of  railways,  insurance  companies,  banks, 
commercial  houses,  and  large  corporations.  Their  use- 


SUMMARY  AND  OUTLOOK  369 

fulness  will  be  greatly  enhanced  when  the  wrong  formulae 
(especially  Formula  1)  now  generally  used  are  replaced 
by  right  ones.1 

But  the  original  purpose  of  index  numbers — to  meas- 
ure the  purchasing  power  of  money  —  will  remain  a 
principal,  if  not  the  principal,  use  of  index  numbers.  It 
is  through  index  numbers  that  we  measure,  and  thereby 
realize,  changes  in  the  value  of  money.  Whether  or  not 
we  ever  stabilize  that  value,  it  is  of  the  greatest  impor- 
tance that  we  know  just  how  stable  or  unstable  our  present 
money  is.  This  is  the  chief  reason  why  today  we  are  so 
much  more  interested  in  index  numbers  than  before  the 
war.  Index  numbers  tell  us  the  value  of  the  mark,  lire, 
and  franc,  at  home  in  terms  of  goods,  as  foreign  exchange 
tells  us  then:  value  abroad  in  terms  of  gold.  And  if,  or 
when,  we  do  regulate  and  stabilize  the  moneys  of  the  world, 
not  simply  relatively  to  each  other  but  relatively  to  goods, 
it  is  the  index  number  which  will  be  requisitioned  to 
measure  and  guide  such  regulation. 


Addendum  to  §9 

Since  this  chapter  was  put  in  paged  type,  the  United  States 
Bureau  of  Labor  Statistics  has  changed  its  system  of  weighting 
by  substituting  the  newly  available  data  of  1919  for  those  of 
1909  hitherto  used.  Their  results  enable  us  to  calculate  the 
index  number  by  Formula  353  between  the  two  years,  1909  and 
1919.  This  turns  out  to  be  1.4  per  cent  lower  than  the 
Bureau's  old  figure  based  on  1909  data  and  that  much  higher 
than  its  new  figure  based  on  1919  data.  The  adjustments 
needed  for  the  intervening  nine  years  barely  exceed  1  per  cent 
in  any  case. 

1  For  a  list  of  current  index  numbers,  see  Appendix  I  (Note  to  Chapter 
XVII,  §  14). 


APPENDIX  I 

NOTES  TO  TEXT 

Note  A  to  Chapter  II,  §  3.  The  Word  "Aggregative."  The  word  "  ag- 
gregative" is  here  proposed  for  general  use  (after  consultation  with  several 
experts)  in  place  of  "price-aggregate"  or  any  other  long  phrase.  I  first 
favored  "aggregatic,"  a  coined  word,  but  Professor  Wesley  C.  Mitchell 
called  my  attention  to  the  existence,  in  the  dictionary,  of  "aggregative." 
Besides  brevity  it  has  several  advantages  over  the  "price-aggregate"  or 
"aggregate-expenditure  method,"  or  other  roundabout  inadequate  phrases 
which  have  been  used,  including  its  applicability  to  quantities,  wages,  etc., 
as  well  as  to  prices. 

Note  B  to  Chapter  II,  §  3.  The  Base  Number  Need  Not  be  100.  Any 
other  number  than  100  may,  of  course,  be  arbitrarily  taken.  As  such  a 
common  base  number,  G.  H.  Knibbs  of  Australia  has  used  1000.  This 
would  change  our  index  number  for  1914  from  the  above  96.32  to  963.2  and 
increase  tenfold  every  other  index  number  in  the  series.  The  London 
Economist  takes  2200  as  the  base  number,  there  being  originally  22  com- 
modities in  the  index  number.  Analogously,  we  could  here  take  3600  as 
the  base  number,  in  which  case  the  index  number  for  1914,  instead  of  the 
above  96.32  would  be  36  times  as  much,  or  the  3467.52  at  the  foot  of  the 
column  in  the  table,  saving  us  the  trouble  of  dividing.  Some  index  num- 
bers take,  as  the  base  number,  the  number  of  dollars  spent  on  a  given  bud- 
get of  commodities  in  the  base  year  or  period.  But,  in  general,  the  100 
per  cent  figure  is  found  most  convenient. 

In  Table  2  in  Chapter  II,  §  6,  while  the  base  number  for  each  individual 
link  is  originally  taken  as  100  per  cent,  in  the  final  series  the  base  numbers 
are  100,  96.32,  97.94,  125.33,  etc.,  the  first  being  used  only  as  base  num- 
ber for  the  second,  the  second  (96.32)  being  likewise  used  only  as  base 
number  for  the  third,  etc. 

Note  to  Chapter  II,  §  11.  Proof  that  for  the  Simple  Geometric,  Fixed  Base 
and  Chain  Methods  Agree.  To  prove  algebraically  the  identity  between 
chain  and  base  averages  under  the  simple  geometric  formula, 


the  1913-1914  link  is  A/—  X  ^  X 

*po      p'o 

and  the  1914^1915  link  is     V~  X  ^  X 


The  chain  index  number  for  1915  relatively  to  1913  via  1914  is  the  product 

371 


372         THE  MAKING  OF  INDEX  NUMBERS 


of  these  two  links ;  and,  in  that  product,  evidently  the  pi's  cancel  out  as 
do  the  p'l's,  etc.,  giving,  as  the  result, 


V—  X  £J  X  .  .  .  which  is  identical  with  the  fixed  base  formula  for  1915 
Po       P  o 

relatively  to  1913. 
Note  to  Chapter  II,  §  13.    Method  of  Finding  the  Simple  Mode.    There  are 


Finding  the  Simple  Mode 


mote 


NUMBER    OF  COMMODITIES 

CHART  62.  Illustrating  the  graphical  distribution  of  the  price  relatives 
and  the  method  of  selecting  the  mode.  (This  chart  is  the  only  one  in  the 
book  not  a  ratio  chart ;  but,  of  course,  the  location  of  the  mode  is  unaffected 
thereby.)  The  top  bar  represents  one  commodity  (coke),  the  price  relative 
of  which  lies  in  the  range  350  to  355  per  cent ;  the  bottom  bar  represents 
two  commodities  (coffee  and  rubber)  in  the  range  80  to  85  per  cent ;  while 
the  mode  occurs  where  there  are  four  commodities — the  largest  number 
—  within  the  range  135  to  140  per  cent. 


APPENDIX  I  373 

many  methods  of  computing  the  mode,  several  graphic  and  several  alge- 
braic. The  method  here  used  is  the  simplest  and  roughest  and  is  illus- 
trated in  Chart  62,  for  prices  for  1917.1  The  largest  price  relative  (351.8) 
lay  between  350  and  355  and  is  represented  by  the  topmost  bar.  The 
smallest  (80.3)  lay  between  80  and  85  and,  as  there  was  another  (83.5)  in 
that  range,  these  two  are  represented  by  the  lowermost  bar  which  is,  there- 
fore, twice  as  long  as  the  uppermost,  or  first  mentioned,  bar.  Between 
these  extremes  are  ranged  the  other  price  relatives  represented  by  the  other 
bars  —  usually  representing  one  price  relative  each  but  in  five  cases, 
including  the  case  of  the  lowermost,  representing  two  price  relatives, 
and  in  two  cases,  representing  four.  The  total  of  the  bars  represents  36, 
the  total  number  of  price  relatives,  or  the  number  of  commodities. 

The  commonest  or  most  frequent  is,  therefore,  the  height  of  one  of  the 
two  fourfold  bars.  The  one  chosen  and  marked  "mode"  has  a  height  of 
135-140.  The  chart  illustrates  the  difficulty  which  often  arises  of  choos- 
ing between  two  squal  frequencies.  Here  the  lower  of  the  two  fourfold 
bars  was  chosen  because,  by  taking  a  range  larger  than  5  points,  the  fre- 
quency within  that  range  is  greater  for  the  neighborhood  of  the  lower  bar 
than  for  that  of  the  upper. 

Note  to  Chapter  II,  §  14.  Proof  that  Fixed  Base  and  Chain  Methods 
Agree  in  Simple  Aggregative.  The  formula  for  the  aggregative  index 
number  for  1915  (year  "2")  relatively  to  1913  (year  "0")  is 

?£*. 

Spo 
On  a  chain  basis,  the  formulae  to  be  multiplied  eventually  are 

the  1913-1914  link  -?£i 

and  the  1914-1915  link 

Spi 

The  chain  index  number  for  1915  relatively  to  1913,  via  1914,  is  the  prod- 

uct of  these  two  links,  i.e.  (after  canceling)  it  is      ^2    which  is  identi- 

Sp0 
cal  with  the  above  formula  for  1915  on  1913  directly  as  base. 

Note  A  to  Chapter  II,  §  15.  The  General  Definition  of  ((  Average."  An  |j 
average,  x,  of  any  series  of  terms,  a,  b,  c,  etc.,  is  any  function  of  these  terms  j 
such  that,  if  they  all  happen  to  be  equal  to  each  other,  x  will  be  equal  to  |  J 
each  of  them  also. 

Thus,  taking  the  simple  arithmetic  average 


where  n  is  the  number  of  terms,  let  us  show  that  this  is  a  true  average  ac- 
cording to  the  definition.  If  each  of  these  terms  happens  to  equal  every 
other,  having  a  common  value,  k,  i.e.  if  a  =  6  =  c  =  .  .  .  =  fc,  then,  evi- 
dently, 

1  This  chart  is  not  a  ratio  chart,  but  the  results  are  not  affected  thereby. 


374         THE  MAKING  OF  INDEX  NUMBERS 

fc       fc       .  .  .       nk       , 


X  =  • 

n  n 


which  was  to  have  been  proved. 

The  simple  harmonic  is  likewise  a  true  average  ;  for,  in  this  case, 


which  was  to  have  been  proved. 
Likewise,  for  the  simple  geometric, 


x  =  :\/abc.  .  "  »  V*  *  *  .  •  •  =  V  kn  =  k. 

Likewise,  as  to  the  simple  median.  For  the  middle  term  of  a,  b,  c,  etc., 
when  they  become  k,  Jfc,  k,  etc.,  is  /c;  and,  as  to  the  simple  mode,  the  com- 
monest term  among  k,  k,  k,  etc.,  is  k. 

As  to  the  simple  aggregative  we  must  start  with  fractions  with  specific 

numerators  and  denominators.     Let  a  be  -j»  b  be  —  >  c  be  —  >  etc.      Then 
the  simple  aggregative  average  of  a,  b,  c,  etc.,  is 

a  + 


A+B  +  C+... 
Ifa  =  6  =  c  =  ...=fc,  then  ~  =  £  =  ^  =  k  and  a  =  kA;    0  =  kB; 

A.        Z5         C/ 

7  =  JfcC,  etc.     Hence,  substituting  these  for  a,  /3,  7,  etc.,  in  the  above 
expression  for  x,  we  have 


which  was  to  have  been  proved. 

We  have  found,  then,  that  all  of  the  six  simple  averages  used  in  this 
book  are  true  averages  according  to  the  definition. 

Weighting  does  not  affect  the  matter;  because  weighting  is,  by  defi- 
nition, merely  counting  a  term  as  though  it  were  two  terms,  or  three  terms, 
or  any  other  number  of  terms. 

The  only  index  numbers  used  in  this  book  which  are  not  true  averages 
are  some,  not. all,  of  the  even. numbered  formulae  (and  derivatives),  which 
are  the'-quotient's  of  a  value  ratio  divided  by  an  average.  Our  definition, 
however,  may  be  modified  to  suit  such  cases  by  specifying  as  the  test  of 
an  average  PLoivthe  price. ratios,  not  only  that  it  shall  equal  the  price  ratios 
if  they  equal  pack-other,  buLaJso  that  at  the  same  time  the  quantity  ratios 
shaft  ejmaleach  otljejc~ 


APPENDIX  I  375 


That  is,  if  P  =         i  -  Q 

Sp0go 

where  Q  is  an  average,  by  the  ordinary  definition,  of  ^,  ^,  etc.,  we  are  to 

go   go 
prove  that  P  =  k  when 

El  =  £J  =  El»  =  .     .  =  ft 

Po       p'o       p"o 

and  when,  at  the  same  time, 


go     go 

The  last  equations  show  that  Q,  being  an  average  of  —,  etc.,  must  be  equal 

So 

to  &'.    Hence 


Since 

it  follows  that  pi  =  fcpo,  etc. 

Substituting  in  the  last  expression  we  have 


which  was  to  have  been  proved. 

It  should  be  noted,  incidentally,  that  the  definition  of  an  average,  as 
originally  stated,  is  a  little  broader  than  that  usually  employed,  which  re- 
quires that  an  average,  to  deserve  the  name,  must  lie  between  the  highest 
and  the  lowest  of  the  terms  averaged.  This  would  rule  out  the  geometric 
average  when  one  of  the  terms  was  zero  or  negative.  But  as  index  num- 
bers are  always  averages  of  positive  terms  this  limitation  of  the  geometric 
does  not  embarrass  us.  Even  other  forms  which,  under  extreme  conditions, 
kick  over  the  traces  seldom  do  so  in  practice. 

Note  B  to  Chapter  II,  §  15.     Proof  that  Geometric  Lies  between  Arithmetic  , 
Above  and  Harmonic  Below.    The  rigorous  proof  of  this  well  known  propo- 
sition  (that  the  geometric  average  necessarily  lies  below  the  arithmetic 
and  above  the  harmonic)  is  to  be  found  in  standard  treatises  on  algebra.1 ' 
But  the  simple  principle  involved  may  be  noted  here. 

Let  us  compare  first  only  the  arithmetic  and  the  geometric  averages  of 
(say)  50  and  200  (the  arithmetic  being  125  and  the  geometric,  100).  The 
geometric  average  is  based  wholly  on  the  idea  of  ratios.  Relatively  to 

1  See,  for  instance,  Chrystal's  Text-Book  of  Algebra,  Part  II,  p.  46. 


376          THE  MAKING  OF  INDEX  NUMBERS 

the  100  the  200  is  twice  as  great  and  the  50  is  "twice  as  small,"  so  that 
"geometrically,"  i.e.  as  to  ratios,  the  two  balance  each  other,  one  being  as 
much  superior  in  ratio  to  100  as  the  other  is  inferior  in  ratio  to  100.  But 
these  equal  ratios  on  either  side  of  100  make  unequal  differences  on  either 
side  of  100 ;  for  the  differences  are  50  below  and  100  above.  Hence  100, 
while  midway  geometrically  between  50  and  200,  is  lower  than  midway 
arithmetically.  Hence  the  arithmetic  average  lies  above  the  geometric. 
Similarly,  the  geometric  average  of  10  and  1000  is  100,  the  1000  being  ten 
times  as  great  as  this  average  and  the  10  being  "ten  times  as  small"  as 
this  average.  But  this  100,  while  half-way  up  from  10  toward  1000  in 
two  equal  ratio  steps,  is  not  nearly  half-way  up  in  two  equal  difference  steps. 
Similarly,  the  geometric  average  of  1  and  10,000  is  100  because  two  steps 
of  one  hundredfold  each  carries  us  from  1  to  10,000,  the  100  being  the  half- 
way step,  but  arithmetically  100  is  far  nearer  to  the  1  than  to  the  10,000. 
r*  In  short,  the  geometric  method  gives  more  influence  to  the  small  magni- 
1  tudes  than  does  the  arithmetic  and  so  results  in  a  smaller  average. 
^^  If  we  take  the  geometric  average  of  any  terms  and  then  take  the  geo- 
metric average  of  their  reciprocals,  these  two  geometric  averages  are  re- 
ciprocals of  each  other.  By  ordinary  algebra  this  is  almost  self-evident,  i.e. 


=  -\/a  X  b  X  c  X  . 
X6  XcX . 


Just  as  the  arithmetic  average  is  necessarily  greater  than  the  geometric 
average  so  the  harmonic  average  is  necessarily  smaller  than  the  geometric 
average. 

This  is  due  to  inverting.  Take  the  same  original  figures,  50  and  200, 
whose  geometric  average  is  100.  Their  reciprocals  are  r&  and  -^  whose 
geometric  average  is  TOTT,  the  reciprocal  of  the  geometric  average  of  the 
original  figures  50  and  200. 

Now  this  inverting  the  three  numbers  50,  100,  200,  has  also  inverted 
their  order  from  the  ascending  order  of  50,  100,  200,  to  the  descending  order 

OI  T&,  Tins')  "SoTT- 

But  the  arithmetic  average  always  lies  above  the  geometric.  The  arith- 
metic average,  therefore,  in  the  series  50,  100,  200,  is  on  the  right  of  the 
100  and  is  on  the  left  of  the  rta  in  the  series,  t^,  -rta,  T^T-  To  be  spe- 
cific, we  may  insert  in  both  cases  the  arithmetic  average  in  parenthesis 
in  its  proper  order,  as  follows  : 

50,  100,  (125),  200,  and  A,  WO,  T*W,  ri* 

Reinverting,  we  obtain  50,  [80],  100,  200  where  the  80  in  square  brackets 
is  the  harmonic  average  of  50  and  200  (i.e.  the  reciprocal  of  the  arithmetic 
average  of  their  reciprocals).  Evidently  this  harmonic  average  is  below 
the  geometric. 

It  is  interesting  to  note  further  that,  when  there  are  only  two  numbers  to 
be  averaged  (a  and  6),  not  only,  as  has  just  been  shown,  does  the  geometric 
average  of  a  and  b  (which  is  Va  X  6)  He  between  the  arithmetic  and  har- 
monic averages  of  a  and  6  but  it  is  their  geometric  average ;  for  the  latter, 
the  geometric  average  of  the  arithmetic  and  harmonic  averages  of  a  and  b, 


APPENDIX  I  377 


after  reduction,  comes  out  Va  X  b,  the  geometric  average  of  a  and  6. 

Note  to  Chapter  III,  §  1.     An  Index  Number  of  Purchasing  Power.     In 
the  book  no  use  is  made  of  the  concept  of  purchasing  power  of  money. 
Everything  which  could  be  said  of  purchasing  power  can  be  said  of  prices 
and  it  may  be  confusing  to  treat  of  both.     An  index  number  of  the  generaf] 
purchasing  power  of  a  dollar  may  be  defined  as  the  reciprocal  of  an  index  I 
number  of  prices.     If  either  is  obtained,  the  other  may  be  obtained  from  ; 
it  by  inversion.     This  index  of  general  purchasing  power  may  also  be  con-^ 
ceived  as  an  average  of  particular  purchasing  powers  over  individual  com- 
modities, each  such  being  defined  as  the  reciprocal  of  a  price,  i.e.  a  "dol- 
lar's worth"  of  anything.     The  ratios  of  such  dollar's  worths  between  any 
two  dates  is  the  reciprocal  of  the  price  relative.     Any  formula  for  prices 
in  this  book  may  be  translated  into  a  formula  of  purchasing  power  by 
substituting  for  p0,  etc.,  the  expression  l/r0,  etc.,  where  r  stands  for  par- 
ticular purchasing  power  and  by  substituting  for  P,  etc.,  the  expression  l/R. 

It  will  be  found  that  a  given  formula  applied  to  work  out  an  index  num- 
ber of  purchasing  power  will  yield  the  same  numerical  result  as  if  applied 
to  work  out  an  index  of  prices  reversed  in  time.  From  this  it  follows  that 
the  reciprocal  of  the  index  number  of  purchasing  power  is  equal  to  the  time 
antithesis  of  the  index  number  of  prices. 

Note  to  Chapter  III,  §  4.  Calculating  the  Weighted  Median  and  Mode. 
According  to  the  definition  of  weights,  a  term  having  a  weight  of  "2"  is 
counted  as  two  terms  ;  and  this  applies  as  readily  to  the  median  as  to  any 
other  average  when  the  weight  is  an  integer. 

When  the  weight  is  not  an  integer  the  same  principle  applies,  though 
not  so  simply.  In  any  case  it  is  well  first  to  arrange  the  price  relatives 
in  order  of  magnitude.  Opposite  such  a  column  we  write,  in  another 
column,  the  weight  for  each  relative. 

This  second  column  has  36  elements.  Their  total  sum,  S,  is  the  total  of 
the  weights.  The  median  is  the  position  in  the  first  column  opposite  the 

c» 

half-way  point  in  the  second.     Take,  then,  half  of  this  sum,  -  .     Then  add 

the  elements  in  the  second  column,  from  above,  till  a  point  is  reached  where 

o 

adding  one  element  more  will  make  the  sum  exceed  -.     Let  A  be  this  sum 

or 

slightly  smaller  than  -.     Proceed  in  the  same  manner  from  below,  obtain- 

Cf 

ing  another  sum,  B,  slightly  smaller  than  -.     Then 


This  leaves  one  middle  element  with  a  weight  which  we  may  call  m,  the 

o 

exceed  -,  and  su 
A  +  m  +  B  =  S. 


o 
element  which  makes  A  or  B  exceed  -,  and  such  that 


378         THE  MAKING  OF  INDEX  NUMBERS 

The  relative  in  the  first  column  opposite  the  weight  m  in  the  second  may 
be  said  to  lie  opposite  the  middle  of  m,  so  that  this  particular  relative  is 
the  required  median  in  case,  and  only  in  case,  half  of  the  second  column 
falls  exactly  at  the  middle  of  m,  i.e.  in  case 


In  all  other  cases,  the  median  is  not  exactly  the  relative  in  the  first  col- 
umn opposite  m,  but  is  an  imaginary  figure  in  the  first  column  above  or 
below  said  relative,  which  imaginary  figure  does  come  opposite  the  middle 
of  S.  This  imaginary  figure  is  interpolated  by  proportional  parts,  i.e.  by 
taking  the  distance  in  the  first  column  between  the  two  neighboring  rela- 
tives between  which  the  median  falls  and  dividing  that  distance  in  the  ratio 
in  which  in  the  second  column,  the  middle  of  S  divides  the  distance  between 
the  middle  of  m  and  the  middle  of  the  neighboring  weight.  (In  practice 
the  operation  is  simplified  by  multiplying  by  two,  i.e.  by  not  halving  the 
two  weights.) 

The  mode  is  calculated  by  the  same  graphic  method  for  the  weighted 
as  for  the  simple  index  number,  i.e.  by  plotting  columns  representing  the  fre- 
quency (or  total  of  weights)  of  price  relatives  which  fall  between  certain 
equidistant  limits,  such  as  100-120,  120-140,  etc.,  and  selecting  the  rela- 
tive having  the  greatest  frequency,  or  highest  column.  Various  devices 
are  resorted  to  to  facilitate  the  work  which  need  not  be  particularized,  as 
the  result  is  always  somewhat  arbitrary  in  any  case. 

Note  to  Chapter  III,  §  7.  Peculiarities  of  the  Aggregative.  It  may  be 
worth  while  here  to  note  that  the  aggregative  is,  in  every  respect,  peculiar 
as  compared  with  the  other  five  types  of  average.  As  we  have  seen,  the 
aggregative  average,  unlike  all  the  other  averages,  is  not  computed  from  the 
mere  price  relatives  or  ratios  of  which  it  is  an  average,  but  requires,  in 
addition,  the  specific  numerators  and  denominators  of  those  ratios  (the 
prices  themselves).  It  follows  that,  if  any  particular  ratio  were  "reduced" 
by  division,  while  that  ratio  itself  would  be  unaffected,  its  numerator  and 
denominator  would  be  affected  and  such  a  change  would,  in  general,  change 
the  resulting  index  number.  For  any  other  type  than  the  aggregative 
it  would  make  no  difference  what  the  numerator  and  denominator  were 
so  long  as  their  ratio  did  not  change. 

Again,  as  we  have  already  seen,  the  simple  aggregative  is  not  simple  in 
precisely  the  same  sense  as  the  other  simple  index  numbers,  because  it 
requires  not  only  the  price  ratios  but  the  prices. 

Finally,  the  weights  used  in  the  aggregative  average  are  not  weights  in 
quite  the  same  sense  as  are  the  weights  used  in  the  other  averages  because 
they  are  applied  not  to  the  terms  averaged  (i.e.  the  price  ratios)  but  to 
their  numerators  and  denominators  separately;  moreover,  these  weights 
are  not  values,  as  are  the  weights  of  the  other  averages,  but  quantities. 

Nevertheless,  the  aggregative  conforms  to  our  general  definition  of  an 
average  given  in  Appendix  I  (Note  A  to  Chapter  II,  §  15)  ;  the  simple 
aggregative  is  analogous  to  the  other  simples  in  that,  given  the  initial  ma- 
terials, in  this  case  the  prices,  they  are  not  reduplicated  but  are  each  taken 
once  only  ;  and,  lastly,  the  weights  used  conform  to  the  general  definition 
of  weights  given  in  Chapter  I,  §  4.  I,  therefore,  prefer  retaining  the 


APPENDIX  I  379 

terms  "average,"  "simple,"  and  "weights"  rather  than  discarding  any 
of  them  in  respect  to  aggregatives. 

Note  to  Chapter  III,  §  11.     Formulas  3  and  17  Reduce  to  58,  and  5  and  19 
to  59,  by  Cancellation.     The  arithmetic  with  weighting  /  (Formula  3)  is 

Pi   i  ^r    t  p'i   I 
~~  Poqo  - — r  P  oq  o  ^-7  ~r  •  •  • 
PO P  o 

; — /   /  _. 

Poqo  +  p  oq  o  +  •  .  • 

Canceling  the  two  p0's  in  the  first  term  of  the  numerator,  and  again  can- 
celing the  two  p'o's  in  the  second,  etc.,  we  have 

+  0'op'i  +  •  •  • 


+  p'oq'o  +  .  .  . 

which  is  identical  with  the  aggregative  with  weighting  7  (Formula  53). 
Similarly,  the  arithmetic  with  weighting  II  (Formula  5)  is 


Po  p'o 


Poqi  +  p'oq'i  +  ... 

which  reduces,  by  canceling  the  p0's,  the  pVs,  etc.,  to 
qipi  +  q'ip'i  +  •  •  • 


which  is  identical  with  aggregative  IV  (Formula  59). 

The  harmonics  III  and  IV  (Formulae  17  and  19)  reduce,  similarly,  to 
the  aggregatives  /  and  IV  (53  and  59)  respectively. 

Note  to  Chapter  III,  §  12.  Professor  Edgeworth's  and  Professor  Young's 
"  Probability"  Systems  of  Weighting  Give  Erratic  Results.  So  far  as  I  know, 
the  only  systematic  methods  of  weighting  not  mentioned  in  the  text  which 
have  been  even  hinted  at  by  other  writers  are  those  mentioned  by  Pro- 
fessor F.  Y.  Edgeworth  and  Professor  Allyn  A.  Young,  modeled  on  proba- 
bility theory. 

Professor  Allyn  A.  Young  proposes,  when  the  data  are  uncertain,  the 
formulae 


and 


and  their  geometric  mean.  His  idea  in  thus  using  squares  of  quantities 
and  values  is  to  follow  the  analogy  of  the  formulas  of  probability  in  the 
method  of  least  squares. 

By  these  formulae,  the  index  number  of  prices  on  1913  as  a  fixed  base 
would  be : 


380 


THE  MAKING  OF  INDEX  NUMBERS 


FORMULA 

1914 

1915 

1916 

1917 

1918 

J  z(goW) 

X  S(?oW) 

101.23 

99.99 

108.36 

149.75 

164.59 

J  s(giW) 

11  2(«iW) 

101.52 

100.52 

108.35 

148.31 

166.68 

Edgeworth,  to  meet  the  case  of  uncertain  data,  proposes  to  use  as  weights 
the  reciprocal  of  the  squares  of  the  deviations  from  some  mean,  on  the 
theory  that  the  price  relatives  which  deviate  the  furthest  are  the  least 
likely  to  be  true  indicators  of  the  general  trend  of  prices  and  therefore 
ought  to  be  given  the  least  weight. 

Edgeworth's  formula  has  not  been  definitely  expressed  and  might  be 
variously  interpreted.  Applied  to  the  geometric  mean  it  may  be  written 


t 

where  d,  d',  etc.,  are  the  percentage  deviations  of  *-lf  ^71,  etc.,  respectively 

Po  p  o 
from  the  mean 


... 

p'o 

By  this  formula  we  would  have  as  our  price  index  number  the  following 


1914 

1915 

1916 

1917 

1918 

96.16 

96.81 

121.24 

165.53 

179.64 

These  results  are  widely  different  from  our  results  by  ordinary  methods. 
Neither  Edgeworth's  nor  Young's  proposed  formula  seems  to  fit  the  case.  I 
agree  with  Walsh  that  ordinarily  it  must  be  presumed  that  the  price  and 
quantity  data  are  not  uncertain  but  certain,  and,  if  certain,  each  has  a  right 
to  be  represented,  not  in  proportion  to  its  deviation  from  some  mean,  but 
in  proportion  to  its  importance  in  the  usual  sense. 

It  seems  to  me  that  the  only  proper  application  of  ideas  of  probability 
to  averaging  price  relatives  is  in  cases  where  the  data  are  actually  defective 
or  uncertain ;  and  the  only  practical  way  in  such  cases  is,  first,  to  write  the 
formula  deemed  best  and  then,  if  the  data  are  considered  as  uncertain,  cor- 
rect this  formula  in  the  individual  cases  of  uncertainty  by  multiplying  by 
arbitrary  coefficients  of  uncertainty. 


APPENDIX  I  381 

At  any  rate,  there  is  ordinarily  no  presumption  that  the  uncertainty  of 
the  data  varies  inversely  as  their  deviation  (or  as  its  square)  from  any  nor- 
mal. Such  a  use  of  the  deviations  might  lead  to  very  bizarre  results. 

Note  to  Chapter  IV,  §  10.  The  Scope  of  Our  Conclusions.  To  see  clearly 
the  formal  framework  of  our  study,  let  us  review  it  briefly.  The  problem 
of  finding  an  index  number  P0i  for  comparing,  on  the  average,  the  prices  (p's) 
of  commodities  at  two  times  was  mathematically  conditioned  by  certain 
p's  and  q's,  the  q's  being  the  coefficients  by  which  the  p's  are  multiplied  to 
give  the  values,  pq's,  of  these  commodities.  So  that  Spi^i  and  Sp0go  are 
the  total  values  of  the  two  groups. 

What  we  have  sought  is  a  formula  or  formulae  for  P0i  such  that,  if  ap- 
plied the  other  way,  PIO,  these  two  applications  will  be  consistent,  i.e. 
PoiPio  will  be  unity,  and  such  also  that  if  the  same  formula  be  applied 
to  the  q's  as  well  as  to  the  p's,  these  two  applications  will  be  consistent, 
i.e.  PoiQoi  =  Sp!gi  -5-  2p<#o  or  PioQio  =  Spotfo  •*•  2pi?i. 

All  our  conclusions  flow  from  the  above  formal  background.  They  are, 
therefore,  of  as  broad  application  as  is  this  background.  They  apply  if 
the  p's  are  wholesale  prices  and  the  q's  are  the  amounts  imported  into  the 
United  States.  They  apply  equally  well  if  the  p's  are  retail  prices  and  the 
q's  are  the  quantities  sold  by  grocers  in  New  York  City.  They  likewise 
apply  if  the  p's  are  rates  of  wages  per  hour  and  the  q's  the  numbers  of  hours 
worked,  or  if  the  p's  are  the  freight  rates  and  the  q's  the  quantities  of  mer- 
chandise transported  from  New  York  to  Liverpool  by  all  Cunard  steamers. 
They  likewise  apply  if  the  p's  are  the  prices  of  industrial  stocks  and  the 
q's  are  the  number  of  shares  sold  by  John  Smith  in  January.  They  like- 
wise apply  to  the  right-hand  side  of  the  "equation  of  exchange." 1  (Some 
critics  have,  because  of  my  interest  in  the  equation  of  exchange,  jumped  to 
the  conclusion  that  my  discussion  of  index  numbers  is  in  some  way  limited 
to  the  problem  of  the  equation  of  exchange !)  They  likewise  apply  if  the 
p's  are  the  lengths  of  the  visiting  cards  of  the  "400"  and  the  q's  their 
breadths,  pq  being  their  area. 

The  results  will  differ  only  when  the  above  mathematical  conditions 
differ.  Thus,  while  we  could  reckon  the  average  change  in  the  length  and 
breadth  of  visiting  cards  between  two  years  so  as  to  preserve  Tests  1  and  2, 
we  would  have  to  modify  our  methods  if  we  were  to  measure  the  average 
change  in  length,  breadth,  and  thickness  of  dry  goods  boxes ;  for  the  en- 
trance of  a  third  factor  in  addition  to  the  two,  p  and  q,  would  change 
the  conditions  of  the  problem.  Likewise  we  would  need  to  modify  our 
methods  if,  for  any  reason,  Tests  1  and  2  are  not  required. 

What  is  emphasized  is  simply  that  within  the  formal  conditions  which   , 
apply  to  the  above  premises  we  find  an  enormous  range  of  problems. 

We  may  formulate  in  the  most  general  way  the  above  mentioned  con- 
ditions to  which  the  reasoning  of  this  book  applies  as  follows : 

Given  a  group  of  variable  magnitudes  which,  under  a  set  of  circum- 
stances designated  by  "  0  "  are  po,  p'o,  p"o,  etc.,  and  which,  under  a  second 
set  of  circumstances  designated  by  "1"  are  pi,  p'i,  p"i,  etc.,  respectively, 
and, 

Given  another  group  of  variable  magnitudes  which  are  in  one  to  one 

1  See  Irving  Fisher,  The  Purchasing  Power  of  Money,  pp.  26,  53,  388,  etc. 


382          THE  MAKING  OF  INDEX  NUMBERS 

correspondence  with  the  members  of  the  first  group,  and  which,  under  the 
eet  of  circumstances  "0,"  are  qo,  q'o,  q"o  and,  under  the  second  set  of  cir- 
cumstances, "1,"  are  qi,  q'i,  q"i,  respectively,  and, 

Given  an  objective  relation  existing  between  the  corresponding  members 
of  the  two  groups  such  that  the  products  paqQ,  p'oq'o,  p"oq"o,  etc.,  on  the 
one  hand,  and  p\q\,  p'iq'i,  p"iq"i>  etc.,  on  the  other,  possess  a  real 
significance  in  the  field  of  study  from  which  the  magnitudes  are  drawn, 
such  that  it  will  be  recognized  as  suitable  for  use  in  checking  up  with  the 
ratios  as  described  below. 

The  problem  is  to  construct  an  index  number 

POI  which  shall  serve  as  a  fair  average  of  £l,  £i  «~i  etc., 

Po  P  o  p    o 

and  Qoi  which  shall  serve  as  a  fair  average  of  —  ,  —  ,  —  *,  etc., 

qo  q'o  q"o 

and  PIO  which  shall  serve  as  a  fair  average  of  _,  —  ,  —  -,  etc., 

Pi  Pi  Pi 

and  Qio  which  shall  serve  as  a  fair  average  of  ^,  ^,  ^—  -,  etc. 

?i  5'i  2"i 

Under  these  circumstances  it  is  fair  to  require  the  fulfillment  of  two 
tests,  viz.,  Test  1  that  P0i  X  PIO  =  1  and  Qoi  X  Qio  =  1  ;  also  Test  2  that 

pol  X  Qoi  =  -^^  and  P10  X  Qio  =  ^M*m    The  justification  of  these 


relations  is  that  they  hold  true  of  the  individual  magnitudes  of  which  POI, 

PIO,  Qoi,  and  Qio  are  averages.     For  example,  we  know  that  El  x  —  —  1 
•*>  po       Pi 

and  £l  X  —  =  2l2l  and  we  can  assign  no  reason  for  violating  the  anal- 

PO       qo       Poqo 

ogous  relationships  among  the  averages,  in  one  direction  rather  than  the 
other. 

With  these  preliminaries,  all  the  reasoning  which  we  have  followed  through 
this  book  applies  whether  the  subject  matter  be  wholesale  prices  and  quan- 
tities marketed,  or  the  length  and  breadth  of  visiting  cards,  or  anything 
whatsoever.  It  certainly  applies  not  only  to  "  general  purpose  index  num- 
bers of  wholesale  prices"  but,  by  merely  using  a  different  set  of  p's  and 
g's,  to  any  special  index  number  of  prices,  say,  of  railway  freight  rates,  or 
to  index  numbers  of  retail  prices,  Raymond  Pearl's  index  number  of  food 
prices  weighted  by  their  calorific  food  values  (the  calories  being,  in  this 
case,  the  q's),  cost  of  living,  wages,  stock  or  bond  prices,  or  sales,  costs  of 
paper  manufacture,  the  rate  of  interest,  «rop  yields,  and  many  others. 

Only  when  the  problem  is  one  which  cannot  be  covered  by  the  above 
formal  and  general  statement  will  our  reasoning  be  inapplicable.  I  know 
no  problem  where  index  numbers  have  yet  been  employed  to  which  these 
general  conditions  do  not  apply.  In  practical  scientific  research,  the 
nearest  approach,  of  which  I  know,  to  such  a  case  as  that  of  the  dry  goods 
box  is  to  be  found  in  anthropometry.  In  comparing  the  shapes  and  sizes 
(i.e.  "  builds  ")  of  two  persons,  or  of  the  same  person  in  two  periods  of 


APPENDIX  I  383 

life,  or  of  two  groups  of  persons,  we  have  such  three  dimensional  problems 
and  their  best  solutions  will,  of  course,  differ  from  those  of  the  two  dimen- 
sional problems  of  this  book. 

Note  to  Chapter  V,  §  2.    Proof  that  the  Product  of  the  Arithmetic  Forward 


by  the  Arithmetic  Backward  Exceeds  Unity.    The  forward  formula  is 

4-}  *(*)*(*) 

and  the  backward,      Pl    so  that  their  product  is     V?D/      Pl     .     We  are 

n  n* 

to  prove  that  this  always  and  necessarily  exceeds  unity.1 

To  prove  this,  we  first  prove  the  more  elementary  theorem  that  the 
simple  arithmetic  average  of  any  number  and  its  reciprocal  exceeds  unity. 
Evidently  one  of  the  two  must  exceed  unity.  Let  1  +  a  be  that  one  and 

let  -  be  the  other.     We  are  to  prove  that 
1  +  a 


On  reducing  and  simplifying,  this  fraction  becomes 


2  +  2a  +  a2 
2  +2a 


1  + 


which  may  be  written 


This  evidently  exceeds  unity,  which  was  to  have  been  proved.    In  other 

words,  the  two  terms,  1  +  a  -\  --  ,  exceed  2. 

l+o 

Applying  the  theorem  just  proved  to  the  problem  in  hand,  we  note  that 
which  is  to  be  multiplied  out,  may  be  written 


(  ^.^2  \—\ 

W    W 


po       p  o       p   o 


.(n  terms)) 
/ 


On  multiplying  these  two  series,  we  see  that  the  product  consists  of  a  series 
of  terms  to  the  number  of  n2.     Some  of  these  terms  (namely,  those  found 

»  It  is  assumed,  of  course,  that  the  price  ratios  ~,  ~,  etc.,  are  not  all  equal  and  that  they 

po  p  o 
are  all  positive  magnitudes. 


384         THE  MAKING  OF  INDEX  NUMBERS 


by  the  vertical  multiplications  such  as  —  X  — ,  etc.)  are  each  evidently 

Po      Pi 
unity.     The  other  terms  may  be  arranged  in  couplets  of  reciprocals  joined 

by  +.     Thus  the  product  of  the  two  factors,  —  X  — -,  may  be  joined  to 

//  Pi      p"o 

its  reciprocal,  —  X  — -,  coming  from  the  same  two  columns.     Since  these 

Po      p"\ 
two  terms  are  reciprocals,  one  must  exceed  unity  and  the  other  be  less  than 

unity,  i.e.  they  may  be  written  (1  +  a)  +  ( ^  which  sum  we  have 

vl  +  a) 

just  shown  exceeds  2.   It  follows  that  the  numerator  of  — \^-L — vPl/_  will 

nz 

have  terms  to  the  number  of  n2,  each  term  being  either  1  or  else  coupled 
with  another  term,  the  two  exceeding  2.  Hence  the  numerator  is  more 
than  n2  while  the  denominator  is  exactly  n2.  Hence  the  whole  fraction 
exceeds  unity. 

Note  to  Chapter  V,  §  6.  The  Two  Steps  between  Weightings  I  and  IV. 
In  the  text,  systems  7  and  77  were  summarily  lumped  together  as  practi- 
cally the  same,  and  likewise  systems  777  and  IV  were  lumped  together. 
Let  us  now  climb  about  from  one  index  number  to  another,  all  based  on 
the  same  price  list  but  varied  by  weighting.  Thus,  in  passing  from  such 
an  index  number  with  the  first  weighting  (7)  to  one  with  the  last  weight- 
ing (IV),  we  shall  take  two  separate  steps,  the  short  step  from  7  to  77  and 
the  long  one  from  77  to  IV,  or,  alternatively,  the  long  one,  7  to  777,  and 
the  short  one,  777  to  IV.  To  fix  our  ideas,  let  us  adopt  the  last  course 
7-777-7F. 

The  first  step  is  the  passage  from  7  to  777,  i.e.  changing  the  weight  of 
bacon  from  p0<7o  to  piq<>  (and  likewise  changing  the  weight  of  barley  from 
p'oq'o  to  p'iq'o,  etc.).  This  change  in  the  weighting  system  has  the  effect, 
as  we  have  seen,  of  loading  the  more  heavily  those  price  relatives  which 
are  already  high,  and,  therefore,  of  raising  considerably  the  index  number 
777  above  7  with  which  we  started.  This  raising  always  happens  whether 
prices  are  rising  or  falling.  That  is,  in  this  first  or  "long"  step  between 
7  and  777,  there  is  no  uncertainty.  Any  index  number  under  the  system 
of  weighting  777  must  be  larger  than  under  weighting  7. 

In  the  "short"  step  between  777  and  IV,  on  the  other  hand,  there  is 
uncertainty.  Any  index  number  under  777  may  be  greater  or  less  than 
under  IV  and  may  even  possibly  happen  under  very  unusual  circumstances 
to  be  much  larger  or  smaller.  It  is  a  fair  lottery.  The  high  price  relatives 
may  draw  either  heavy  weights  or  light  weights  with  an  even  chance  each 
way,  as,  likewise,  may  the  low  price  relatives.  The  net  effect  will  prob- 
ably be  an  almost  complete  offsetting  so  that  the  final  index  number  (IV) 
will  probably  be  close  to  777  and  may  be  either  slightly  above  or  below.1 

1  It  is,  of  course,  conceivable  that  there  is  a  correlation  between  the  prices  and  quan- 
tities but  this  may  be  in  either  direction  according  as  the  prime  mover  is  supply  or  demand. 
In  the  case  in  hand  there  is  essentially  no  correlation  and  investigation  of  some  New  York 
Stock  Exchange  prices  shows  the  same  absence  of  correlation.  In  the  case  of  the  12  crops 
used  by  Day  and  Persons  of  the  Harvard  Committee  on  Economic  Research,  where  supply 


APPENDIX  I  385 

So  that  after  both  steps  are  taken,  and  we  compare  I  with  IV,  we  cannot, 
as  we  could  in  the  case  of  type  bias,  be  absolutely  sure  of  the  result.  All 
that  we  can  say  is  that  it  is  exceedingly  probable  that  IV  will  exceed  7. 
No  case  to  the  contrary  occurs  in  the  present  investigation  and  it  seems 
very  unlikely  that  such  a  case  will  ever  be  encountered  in  practice  (except 
for  the  mode  and,  in  rare  cases,  the  median,  or  except  when  there  are  a 
very  few  commodities  in  the  index  number). 

But  we  have  not,  even  yet,  thrown  our  results  respecting  weight  bias 
into  a  form  quite  comparable  with  that  employed  for  type  bias.  In  taking 
each  of  the  two  steps,  the  "long"  and  the  "short,"  we  have  used  only 
forward  index  numbers.  But  now,  after  putting  the  two  steps  together, 
we  are  ready  to  revert  to  the  original  method,  that  of  multiplying  together 
forward  and  backward  index  numbers  of  the  same  kind. 

Thus,  we  have  shown  that  (in  all  probability)  geometric  IV  forward  is 
always  a  larger  index  number  than  its  time  antithesis,  geometric  7  forward. 
But  geometric  7  forward  is  the  reciprocal  of  geometric  IV  backward.  In 
proof :  Geometric  7  (23)  forward  is 


2po?o/  (pi  \poqo       (p\\p'oq'o 
V  W  C  W 


Geometric  IV  (29)  forward  is 

Spitf:  ~* 


Geometric  IV  backward  is  found  by  interchanging  the  "O's"  and  "IV 
in  geometric  IV  forward  and  is 


I/  X  ...  3 

Evidently  the  first  of  these  three  formulae  (geometric  7  forward)  is  the 
reciprocal  of  the  last  (geometric  IV  backward),  which  was  to  have  been 
proved.  It  follows  that  the  product  of  geometric  IV  forward  X  geometric 
IV  backward  is  the  same  as 


geometric  IV  forward  X  f  • 


^geometric  7  forward/ 
which  is 

geometric  IV 
geometric  7 

and  this  (since  IV  always  exceeds  7)  is  greater  than  unity.  Consequently 
the  original  product,  geometric  IV  forward  X  geometric  IV  backward, 
exceeds  unity.  In  other  words,  they  have  a  positive  joint  error.  In  still 

was  the  dominant  variable  in  changing  the  quantities  marketed  and  where  there  is  an  in- 
verse correlation  between  quantity  and  price,  weighting  system  /  makes  for  a  higher  index 
number  than  II,  and  ///  than  IV.  Yet  it  is  noteworthy  that  the  effect  on  the  curves  given 
in  Chapter  XI  is  almost  negligible. 


386         THE  MAKING  OF  INDEX  NUMBERS 

other  words,  geometric  IV  has  an  upward  bias,  and  this  bias  is  acquired 
through  weighting,  in  exactly  the  same  sense  as  type  bias  previously  found 
for  the  arithmetic  and  harmonic. 

Likewise,  we  could  trace  the  transformation  of  an  index  number  by  chang- 
ing its  system  of  weighting  from  //  to  ///  vial  or  IV,  i.e.  first  by  the  short 
step-  from  //  to  /  and  then  the  long  one  from  /  to  ///,  or  first  by  the  long 
step  from  II  to  IV  and  then  the  short  step  from  IV  to  ///.  In  all  cases, 
changing  ar  quantity  element  in  the  weights  has  only  a  small  effect,  which 
must,  in  general,  be  assumed  equally  likely  to  be  in  either  direction  ;  but 
a  change  of  the  price  element  in  the  weight  has  a  larger  effect  and  in  a 
definite  direction. 

By  such  reasoning  we  may  impute  upward  bias  to  geometric  IV,  geo- 
metric III,  median  IV,  median  ///,  mode  IV,  mode  ///,  while  similarly, 
/  and  II  have  a  downward  bias  for  the  same  three  types  (geometric,  me- 
dian, mode). 

The  arithmetic  and  the  harmonic  remain.    We  are  to  show  that,  for 
instance,  arithmetic  IV  forward  X  arithmetic  IV  backward  exceeds  unity. 
In  algebraic  terms,  in  the  first  place,  the  arithmetic  IV  backward  is  the 
same  as  the  reciprocal  of  harmonic  /  forward. 
For 

arithmetic  IV  forward  is 


arithmetic  IV  backward  is 


Its  reciprocal  is 


This  is  harmonic  /,  which  was  to  have  been  proved. 
Hence  (to  retain  for  comparison  the  "spelled  out"  method  just  em- 
ployed), 

arithmetic  IV  forward  X  arithmetic  IV  backward 

is  arithmetic  IV  forward  X  f 

«tTC4-lfrvtsvti 


harmonic  7  forward- 
arithmetic  IV  forward 


harmonic  /  forward 

That  this  exceeds  unity,  or  that  the  numerator  exceeds  the  denominator, 
remains  to  be  proved.  We  shall  see  that,  not  only  does  the  numerator 
exceed  the  denominator,  but  that  the  numerator  exceeds  arithmetic  /, 


APPENDIX  I  387 

or  arithmetic  77,  or  harmonic  777,  or  harmonic  IV,  and  that  these  exceed 
the  denominator. 

In  the  first  place,  arithmetic  IV  exceeds  arithmetic  77  as  we  have  al- 
ready proved  with  certainty  and,  since  arithmetic  7  in  all  probability 
agrees  closely  with  arithmetic  77,  it  follows  that,  in  all  probability,  arith- 
metic IV  exceeds  arithmetic  7.  But  we  have  seen  that  arithmetic  7  is 
identical  with  harmonic  777  (both  being  Laspeyres')  while  arithmetic  77 
and  harmonic  IV  are  both  Paasche's.  And  we  know  that  harmonic  777 
exceeds  with  absolute  certainty  harmonic  7.  Thus,  the  numerator  is 
greater  than  and  the  denominator  is  less  than  (indifferently)  arithmetic 
7  and  77,  or  harmonic  777  and  IV,  which  was  to  have  been  proved. 

Note  to  Chapter  V,  §  9.  Formula  9  after  Reversing  Subscripts  and  In- 
verting Becomes  18.  Formula  9  (or  arithmetic  IV)  forward  being 

Sptfi^-1 
2pi0i 

its  backward  application,  found  and  represented  by  the  dotted  line  in 
Charts  18P  and  ISQ  by  reversing  the  subscripts,  is,  as  shown  in  the  last 
note, 


the  reciprocal  of  which  (represented  graphically  by  prolonging  the  dotted 
line  in  Charts  18P  and  18Q)  is 


But  this  is  Formula  13  (or  harmonic  7),  which  was  to  have  been  proved. 

Note  to  Chapter  V,  §  11.  Bias  and  Dispersion  in  Formula.  Any  bias, 
as  has  been  seen,  is  defined  in  terms  of  some  joint  error.  Thus  the  joint 
error  (in  this  case  joint  bias  (B))  of  the  arithmetic  forward  and  backward 
is  given  by  the  formula  1  +  B  =  arithmetic  forward  X  arithmetic  back- 
ward, or,  calling  arithmetic  forward  A  and  remembering  that  arithmetic 
backward  is  the  reciprocal  of. the  harmonic  forward  (which  we  may  call 

H),  we   have  1  +  5  <=  A  X  —  =4-    But  tne  bias  of  the  arithmetic  for- 
77      H 

ward  is  not  the  whole  of  B  since  1+5  expresses  the  full  ratio  of  the  up- 
ward biased  A  to  the  downward  biased  H  and  so  involves  a  double  appli- 
cation of  bias.  Thus,  we  may  define  the  bias,  b,  as  half  of  B,  or  rather 
half  geometrically  (as  in  compound  interest)  according  to  the  formula 

(!+&)«-!+  5, 


388         THE  MAKING  OF  INDEX  NUMBERS 

whence1 


VAR 

Our  main  formulae,  then,  are 

A 


H 


either  of  which  may  be  derived  from  the  other,  the  fprmer  expressing  the 
upward  bias  of  A  relatively  to  VAH,  and  the  latter  expressing  the  down- 
ward bias  of  H  relatively  to  VAH. 

We  next  need  a  "dispersion"  index,  d,  to  represent  the  degree  of  the 
divergences  of  the  price  ratios  from  each  other.  Let  us  begin  with  the 
case  of  only  two  commodities,  considered  of  equal  importance,  their  price 
relatives  (or  quantity  relatives,  or  whatever  the  subject  matter  may  be) 
being  r  and  r',  where  r  is  the  larger.  The  total  divergence  from  each  other, 
D,  of  r  and  r'  may  be  defined  by  1  +  D  =  r/r'.  But  a  preferable  magni- 
tude to  use  is  not  this  total  divergence  between  the  two  but  the  average 
dispersion,  d,  from  a  common  mean,  best  taken  as  their  geometric  mean  so 
that  d  is  half  of  D  geometrically,  i.e. 


— t* 
Hence 


1  F_/r'    _Vrr7 

1+  d     \r    V^'      r  V 


From  these  equations  we  may  derive 
r  - 


Since  there  are  but  two  relatives  to  be  averaged,  r  and  r',  their  simple 

< 

1  In  the  same  way  beginning  with  the  harmonic,  instead  of  the  arithmetic,  and  using  6' 

rr 

to  express  a  downward  bias  we  could  derive  1—6'  =     .  —  •..    But  since  by  multiplying 

v  AH 
together  the  equations  for  1  +  6  and  1  —  6'  we  get  (1  +  6)(1  -  60  -  1  we  can  better 

use,  instead  of  1  -  6',  the  equal  expression,    ,  .and  dispense  with  the  use  of  6'  altogether. 

l+o 


APPENDIX  I  389 

i  r/ 
arithmetic  average  (A)  is  A  =  — — —  and  their  simple  harmonic  average 

(H )  is  H  =  — - — .    In  these,  we  substitute  the  above  expressions  for 

r 
r  and  r'  giving 


r+r- 


A  =  -±- 
2 

H         2 


1     _l         1      +  n  +  d) 
whence,  dividing  the  equations  and  canceling  Vrr7, 

#~ 

In  other  words,  the  result  is  independent  of  the  actual  magnitude  of  the 
price  relatives  and  dependent  only  on  the  ratio  (1  +  d)  of  their  divergence 
from  their  mean.  Anticipating  this  result,  we  might  have  substituted 
for  the  above  proof  the  following  simplification : 

Let  the  (geometric)  average  of  the  two  price  relatives  be  considered  as 

100  per  cent,  or  unity,  the  upper  one,  1  +  d,  and  the  lower, .    Then 

1  +  d 


and 


+ 


U-M) 

whence  evidently  —  = 

H 

as  before. 

But  we  already  know  that  (1  -f  &)2  is  also  equal  to  — .    Hence  we  have 

H 

(after  extracting  the  square  roots) 


390         THE  MAKING  OF  INDEX  NUMBERS 

as  the  equation  expressing  the  relation  between  the  bias  b  and  the  disper- 
sion index  d. 

That  bias  and  dispersion  are  both  relative  to  the  same  axis  or  mean  pro- 
portional can  readily  be  shown  in  several  ways  from  the  above  equations. 
The  mean  proportional  with  reference  to  which  6  was  reckoned  was  VAH 
and  that  with  reference  to  which  d  was  measured  was  Vrr',  and  these  two 
expressions  are  readily  shown  to  be  equal. 

From  this  formula  it  will  be  seen  that  the  bias  increases  very  rapidly 
with  an  increase  in  the  dispersion,  that  when  the  dispersion  is  zero  the 
bias  is  zero,  when  the  dispersion  is  5  per  cent  the  bias  is  negligible,  when 
the  dispersion  is  50  per  cent  the  bias  is  8.34  per  cent  as  the  following  table 
shows : 

TABLE  48.   FOR  FINDING  THE  BIAS  CORRESPOND- 
ING TO  ANY  GIVEN   DISPERSION 

(Both  in  per  cents) 


DISPERSION  (d) 

BIAS   (6) 

5 

.12 

10 

.45 

20 

1.67 

30 

3.46 

40 

5.72 

50 

8.34 

100 

25.00 

So  much  for  the  simple  case  of  two  price  relatives  and  where  the  disper- 
sion is  self-evident.  Where  there  are  more  than  two  the  dispersion  must 
be  some  sort  of  average.  To  obtain  such  an  average,  we  substitute,  in 
thought,  two  imaginary  price  relatives  for  all  the  actual  ones,  the  disper- 
sion of  each  of  these  two  from  their  mean  being  an  average  of  all  the  actual 
deviations  of  the  36  from  their  mean.  Various  such  averages  have  been 
used  to  measure  dispersion.  That  usually  employed  is  the  "standard 
deviation"  obtained  by  taking  the  average  of  the  squares  of  the  deviations 
of  the  individual  price  relatives  (each  deviation  being  measured  from  the 
arithmetic  average)  and  extracting  the  square  root.  Another  is  analogous 
to  the  above  but  is  geometric  in  nature  instead  of  arithmetic.  It  is  found 
by  taking  the  standard  deviation  of  the  logarithms  of  the  price  relatives 
and  then  taking  the  anti-logarithm  of  that.  Another  is  the  average 
"spread"  between  the  median  and  the  two  "quartiles." 

Of  these  the  middle  one  seems  the  best  adapted  to  the  present  purpose. 
It  is  certainly  better  adapted  theoretically  than  the  first  (the  ordinary 
arithmetically  defined  standard  deviation),  because  the  price  relatives  and 
quantity  relatives  with  which  we  have  to  deal  are  widely  varying  and  have 
"skew"  distribution  varying  more  upward  than  downward  which  the 
geometric  or  logarithmic  standard  deviation  tends  to  eliminate. 


APPENDIX  I 


391 


Practically,  however,  the  arithmetic  and  geometric  standard  deviations 
agree  surprisingly  well  in  spite  of  the  skewness  and  greatness  of  the  dis- 
persion. This  will  be  seen  from  the  following  table : 


TABLE  49.   STANDARD  DEVIATIONS   (FOR  PRICES) 
(In  per  cents) 


FIXED  BASE 

1914 

1915 

1916 

1917 

1918 

Arithmetic  S.  D  

10 

16 

24 

58 

45 

Geometric  S.  D  

11 

17 

21 

39 

33 

CHAIN  OF  BASES 

Arithmetic  S.  D  

10 

12 

27 

29 

20 

Geometric  S.  D  

11 

12 

22 

22 

22 

We  may  then  picture  the  dispersing  terms  (price  relatives,  or  quan- 
tity relatives,  or  whatever  the  terms  under  consideration  may  be)  as  all 
reduced  to  two  imaginary  terms,  say  price  relatives,  one  lying  above  the 
(geometric)  average  and  representing  all  the  actual  price  relatives  above 
the  average,  and  the  other  lying  below  that  average  and  representing  all 
the  actual  price  relatives  below  the  average,  and  each  diverging  from  that 
average  in  the  ratio  1  +  d  (and  from  each  other  in  the  ratio  (1  +  d)2)-  In 
this  empirical  way  we  reduce  the  complex  case  of  many  price  relatives 
to  the  original  and  simpler  case  of  only  two  price  relatives. 

The  question  now  arises :  Will  the  dispersion  index  d  as  thus  denned 
(i.e.  as  the  geometrically,  or  logarithmically,  determined  standard  devia- 
tion) be  actually  related  to  the  bias  b  according  to  the  formula  1  +  6  = 

TTd 

^—  which  we  found  to  be  true  in  the  simple  two  term  case? 


The  answer  is,  yes,  very  closely. 

First  we  shall  show  that  the  above  empirical  relation  between  the  bias  b 
and  the  dispersion  index  d  can  be  made  absolutely  exact  for  the  case  of  any 
number  of  commodities  if  we  suitably  change  the  definition  of  d  to  a  fourth 
form,  in  terms  of  A  and  H,  as  follows : 

To  maintain  absolutely  the  equation 


l+d 


392         THE  MAKING  OF  INDEX  NUMBERS 
we  simply  use  it  and  the  equation 

\~A  A 

/*i     i    l*\  IJ^  "*^ 

from  which  to  derive 


(!+<*) 


U+d) 


fA 
VAH' 

Solving  this  quadratic  equation  f or  1  +  d  and  reducing  we  have 


This  new  determination  of  d  is  relative  to  VAH  as  before. 

The  above  formulae  will  serve  also  for  the  harmonic  except  that  whereas 

1  +  d  is  the  magnitude  pertaining  to  the  arithmetic,  is  the  magni- 

1  +d 

tude  pertaining  to  the  harmonic,  the  d  being  the  same. 

It  only  remains  to  show  that  this  special  form  of  dispersion  index  (in 
terms  of  A  and  H  and  therefore  also,  of  course,  in  terms  of  the  original 
data  themselves)  is,  in  actual  fact,  very  close  to  the  geometric  (logarith- 
mically calculated)  standard  deviation,  as  the  following  figures  show : 

TABLE  50.  SPECIAL  DISPERSION  INDEX  COMPARED 
WITH  STANDARD  DEVIATION  (LOGARITHMICALLY 
CALCULATED)  FOR  THE  36  COMMODITIES  (SIMPLE) 

(In  per  cents) 


SPECIAL 

STANDARD 

1914      

11.5 

11.5 

1916      

173 

17.2 

1916     

21.5 

21.4 

1917     

39.2 

38.7 

1918     

33  7 

33  1 

For  the  weighted  arithmetic  and  harmonic  the  case  is  only  slightly  dif- 
ferent.   We  then  have,  for  instance, 


1  +d 


where  A'  and  H'  are  weighted  arithmetic  and  harmonic  index  numbers 
whence 


APPENDIX  I 


393 


-  A'H'  +Af 


This  also  is  close  to  the  (logarithmically  calculated)  standard  deviation  as 
the  following  table  (in  which  the  weighted  averages  have  the  mean  weights 
Vpotfopitfi,  etc.,  as  per  Formulae  1003  and  1013)  shows : 

TABLE  51.  SPECIAL  DISPERSION  INDEX  COMPARED  WITH 
STANDARD  DEVIATION  (LOGARITHMICALLY  CALCU- 
LATED) FOR  THE  36  COMMODITIES  (WEIGHTED) 

(In  per  cents) 


SPECIAL 

STANDARD 

1914  

8.3 

7.7 

1915  

15.3 

15.1 

1916  

19.2 

19.2 

1917  

39.1 

38.9 

1918  

26.2 

26.5 

So  much  for  the  type  bias  as  applied  to  the  simple  arithmetic  or  har- 
monic, and  as  applied  to  their  mean  weighted  forms.  We  have  still  to 
consider  the  weight  bias  of  the  various  systems  of  weighting. 

Summarizing  the  proof  in  its  simplest  form,  let  us  assume  only  two  com- 
modities as  before,  their  price  relatives  (—  and  £-*  ]  being  1  -\- d  and 

VPo  p'oj 

.     As  to  the  weights  p^qo,  p'aq'o  and  p\q\,  p'lq'i,  we  may  call  po  and 

l+o 

p'o  100  per  cent,  or  1,  so  that  pi  and  p\  are  1  -f  d  and  ,  while  as 

1  +  d 

to  the  quantities  we  assume  they  do  not  change,  i.e.  q0  =  q\  and  q'o  =  q'i 
(which  may  be  called  q  and  q'  simply)  and  that  they  are  such  as  to  make 
equal  the  average  weights  of  the  two  price  relatives  over  the  two  years, 

Substituting  in  this  equation  the  above  values  for  the  p's,  viz.,  po  ==  1, 
pi  =  1  +  d,  and  p'0  =  1,  p'i  =r-T-^f 


v  (1 


-M) 


i.e.  (remembering  the  above  q  equalities,  qQ  =  qi  and  q'0  *  q'i), 

'/ 


394         THE  MAKING  OF  INDEX  NUMBERS 


whence,  (1  +  d)g2  =  --,  or  (1  +  d)V  =  g'2,  or  (1  +  d)2  =  -,or      = 

1  +  d  T        q 

I  +  d  or,  letting  #  =  1,  simply,  q1  -  1  +  d. 

Summarizing,  we  may  now  substitute,  in  any  formula  to  be  investigated, 
the  following  magnitudes  :    p0  =  1,  p'o  =  1,  So  =  1,  g'o  =  1  +  d,  p\  = 


Applying  these,  we  find  that  Formulae  53,  54,  353,  123,  125,  323,  325 

(some  of  which  have  not  yet  been  explained)  reduce  to  unity  so  that  we 
may  consider  the  bias  of  the  formulas  to  be  investigated  as  measured  rela- 
tively to  any  one  of  these  as  a  basis.  The  bias  of  any  formula  becomes 
simply  the  value  of  that  formula  after  substituting  the  above  eight  values 
for  po,  p'o,  qo,  q'o,  Pi,  P'I,  q\,  q'i- 

The  following  are  the  results  for  index  numbers  by  Formulae  1003,  7 
or  9,  27  or  29. 

1  +  d  -\ 


™ 
1003  l+o  --  d  whence  &  =  2  (1  +  d) 

7or9    l+6  =  l+d+  —  --  lwhenceo  =  -^—  (2) 

1  +  a  1  +  d 


27  or  29        1  +  6  =  (1  +  d)*  +  <*      whence  b  =  -£—  +  ...       (3) 

the  terms  omitted  in  the  last  being  negligible. 
Equation  (1)  gives  the  bias  of  the  singly  biased  arithmetic 

and  of  the  singly  biased  harmonic. 

Equation  (2)  gives  the  bias  of  the  doubly  biased  arithmetic 
and  of  the  doubly  biased  harmonic. 

Equation  (3)  gives  the  bias  of  the  singly  biased  geometric. 
The  equations  are  given  in  terms  of  upward  bias  but  the  corresponding 
downward  biases  also  (i.e.  of  Formulae  1013  ;   13  and  15  ;   23  and  25)  are  im- 

plicitly given  merely  by  inverting,  i.e.  taking  -  . 

l+o 

Evidently  (as  equation  (2)  shows)  Formula  9,  or  Palgrave's  formula,  has 
a  double  dose  of  upward  bias  as  compared  with  the  bias  (shown  by  equa- 
tion (1))  for  1003,  the  mean  weight  arithmetic.  That  is,  besides  the  type 
bias,  which  Formula  1003  has,  there  is  the  weight  bias  of  9  and  the  one  is 
equal  to  the  other.  The  weight  bias  (given  by  equation  3)  of  the 
geometric,  Formula  29,  is  evidently  larger  than  either  of  the  (single) 
biases  as  given  in  the  first  two  equations.  It  is  larger  than  the  first,  both 
because  its  denominator  is  less  by  d  and  because  there  are  other  terms  to 
be  added,  although  d  is  so  small  compared  with  2  and  with  2  +  d  and  the 
additive  terms  are  also  so  small,  each  involving  a  power  of  d,  that  the 
entire  difference  between  the  last  equation  and  the  first  is  negligible. 

The  above  equations  are  not  only  absolutely  true  under  the  special  con- 


APPENDIX  I 


395 


ditions  assumed  but  are  approximately  true  in  actual  cases  such  as  that 
of  the  36  commodities.  The  dissimilarity  between  the  equations  for  the  bias 
of  the  arithmetic  and  harmonic  index  numbers  (1003  and  1013)  and  that 
for  the  weighted  geometries  (23,  25,  27,  29)  might  lead  one  to  suppose  that 
they  would  give  widely  different  results.  But  when  we  calculate  them  we 
find  they  agree  almost  exactly,  as  the  following  table  shows,  giving  the 
bias  (6)  of  both  corresponding  to  various  standard  deviations  (d). 


d 

6 

* 

ARITHMETIC 
HARMONIC 
(1003,  1013) 

GEOMETRIC 
(23,  25,  27, 
29) 

5 

.12 

.12 

10 

.45 

.45 

20 

1.67 

1.67 

30 

3.46 

3.48 

40 

5.72 

5.77 

50 

8.34 

8.45 

200 

25.00 

25.99 

We  could,  of  course,  make  the  equations  absolutely  exact  by  suitably 
adapting  the  definition  of  dispersion  to  each  particular  case.  But  the  object 
of  this  Appendix  note  has  been  to  show  how  the  size  of  the  bias  is  related 
to  the  size  of  the  dispersion  of  the  original  data.  Where  there  is  only 
slight  dispersion  the  error  caused  by  using  a  biased  formula  is  small  but  as 
the  dispersion  increases  the  error  thus  introduced  increases,  and  in  a  much 
faster  ratio.  Consequently,  in  cases  of  wide  dispersion,  such  as  those  of 
the  36  commodities  (for  1917  relatively  to  1913),  the  upward  bias  of  For- 
mula 1,  for  instance,  or  the  downward  bias  of  23,  is  very  great. 

For  any  particular  set  of  statistics  we  can,  by  calculating  the  standard 
deviation  or  dispersion  index,  and  from  it  the  bias  of  any  biased  formula, 
tell  in  advance  whether  the  use  of  that  formula  will  introduce  too  large  an 
error  to  make  its  use  permissible.1 

Note  to  Chapter  VI,  §  1.  //  One  Formula  is  the  Time  Antithesis  of 
Another,  the  "Other"  is  of  the  "One."  This  is  very  simply  shown.  Let 
Poi  stand  for  any  index  number,  taken  forward,  i.e.  for  time  "1"  relatively 
to  time  "0. "  Our  twofold  procedure  gives : 

Starting  with  P0i 

(1)  By  reversing  the  times,  PIO 

(2)  By  inverting  the  last,— 

PIO 
which,  therefore,  is  the  time  antithesis  of  the  original  Pw.     We  are  to  show 


1  See,  for  instance,  Chapter  XVI,  §  6,  for  discussion  as  to  the  large  bias  in  Sauerbeck's 
index  numbers. 


396         THE  MAKING  OF  INDEX  NUMBERS 

that  starting  with  the  last  formula  and  applying  the  same  twofold  pro- 
cedure we  shall  reach,  as  its  time  antithesis,  the  original  formula. 

Starting,  then,  with  — 
Pio 

(1)  By  reversing  the  times,  — - 

Poi 

(2)  By  inverting  the  last,  Poi 
which  was  to  have  been  found. 

Note  to  Chapter  VII,  §6.  The  Cross  between  Two  Factor  Antitheses 
Fulfills  Test  2.  Discussion.  Let  POI  be  any  given  formula.  Its  factor  an- 
tithesis is  ??1^1  -s-  Qoi  where  Qoi  is,  of  course,  the  formula  corresponding 

Sp<#o 
to  POI  applied  to  quantities.    Their  cross  or  geometric  average  is 


This  last  formula  fulfills  Test  2  because  its  factor  antithesis  is,  inter- 
changing p's  and  q'a, 


and  this,  multiplied  by  the  preceding,  gives    ^*,  as  the  test  requires. 


We  have  considered  the  rectified  formula  for  prices  a  cross  between  the 

original  formula  P0i  and  its  time  antithesis,  —  Hlii  -±-  Q01. 

2po?o 
But,  evidently,  the  same  expression  may  be  written  more  symmetrically  : 


1   NX    ^   IP01 


'Zpoqo*    VQ* 


while,  likewise,  the  rectification  of  Qoi  is 


In  these  forms  for  the  rectified  formulae  the  two  factors  are  not  index 
numbers.  The  first  factor,  in  both  cases,  is  the  mean  between  the  value 
ratio  and  unity,  or  100  per  cent.  Thus,  if  the  value  ratio  is  121  per  cent, 
its  square  root,  or  the  mean  between  it  and  100  per  cent,  is  1 10  per  cent.  This 
is  what  each  index  number,  that  for  prices  and  that  for  quantities,  would 
be  if  they  were  equal ;  that  is,  it  is  their  geometric  mean  or  average. 

The  other  factor,  in  each  case,  is  the  multiplier  or  corrector  of  that  aver- 
age, which  is  necessary,  in  the  one  case,  to  produce  the  rectified  price  index, 
and,  in  the  other,  to  produce  the  rectified  quantity  index.  These  two 
factors  are  reciprocals  of  each  other,  one  magnifying  and  the  other 
reducing  the  average  in  a  certain  proportion.  Thus  if  POI  is  two  per  cent 
greater  than  QM,  this  two  per  cent  is  apportioned  equally  on  both  sides  of 


APPENDIX  I  397 


the  mean,  110,  —  the  rectified  P  being  110  X  (or  about  one  per  cent 

above  110)  and  the  rectified  Q  being  110  X  Vi$f  (or  about  one  per  cent 
below  110). 

The  first  factor  -y/^Ml  might  be  called  the  half-way  ratio,  being  at 

\2po<?o 

once  the  mean  between  100  per  cent  and  the  value  ratio  and  also  between 
the  rectified  P  and  Q  (or  unrectified,  for  that  matter)  while  the  second  factor 

A/—  —  or  \/—  might  be  called  the  price  multiplier  or  quantity  multiplier. 
*  Qoi  ^  POI 

In  these  terms  we  may  say  that  the  rectified  index  numbers  of  prices 
and  quantities  are  each  obtained  from  the  half-way  ratio  by  means  of  price 
and  quantity  multipliers. 

The  reader  may  be  interested  in  following  through  the  application  of 
the  preceding  remarks  to  the  rectification  of  Formula  3  (which  is  the  same 
as  of  4,  5,  6,  17,  18,  19,  20,  53,  54,  59,  or  60),  the  results  of  which  are  very 
simple. 

Thus,  for  prices,  the  result  is 


i  x 


The  four  magnitudes  entering  into  this  expression  are,  of  course,  the 
same  as  those  entering  into  that  already  given  for  103P  and  103Q.  By 
merely  a  change  in  the  order  four  different  formulae  are  formed,  two  for 
103P  and  two  for  103Q. 

Note  A  to  Chapter  VII,  §  8.    Given  Two  Time  Antitheses,  Their  Respective 

Factor  Antitheses  are  Time  Antitheses  of  Each  Other.    Let  P0i  and  -  i-  be 

PIO 

any  time  antitheses  and  let  Qoi  and  —  -  -  (that  is,  the  same  formulae  applied 

Qio 

to  quantities)  likewise  be  time  antitheses  of  each  other.    Then  the  factor 
antitheses  of  the  first  two  are 


-. 

Qio 

These  are  evidently  time  antitheses  of  each  other  because  by  interchanging 
the  "O's"  and  "1's"  of  either  formula  and  then  inverting,  we  turn  each 
into  the  other. 

Note  B  to  Chapter  VII,  §  8.     Given   Two  Factor  Antitheses,  Their  Re- 
spective  Time  Antitheses  are  Factor  Antitheses  of  Each  Other.     Let  P0i  and 

^>1^1  -r  Qoii  be  any  two  factor  antitheses.    Evidently  their  respective 


time  antitheses,  viz.  •—  -  and  Qw  -fr  -£522,  are  also  factor  antitheses  of 

Pio 
each  other. 


398        THE  MAKING  OF  INDEX  NUMBERS 

Note  to  Chapter  VII,  §  9.  Rectification  May  be  First  of  Time  Antitheses 
and  then  of  Factor  Antitheses,  or  Vice  Versa,  or  Simultaneously.  In  general 
terms  any  quartet  of  formulae  is 

P  l 

Poi  — 


Spogo  Spogo 

Qoi  1 

The  two  crosses  of  time  antitheses  are 

(1) 


(2) 


the  latter,  (2),  of  which  reduces  to 


which  is  the  factor  antithesis  of  the  former,  (1),  being  obtainable  from 
it  by  interchanging  the  p's  and  g's  and  dividing  into 

The  two  crosses  of  factor  antitheses  are 


V 


PoiX    SPogo  (3) 

\Qoi/ 


These  are  time  antitheses  of  each  other ;  if  in  either  we  reverse  0  and  1  and 
invert  we  get  the  other. 

Inspection  will  also  show  that  the  cross  of  either  of  the  above  pairs  of 
crosses  as  well  as  the  fourth  root  of  the  product  of  the  original  quartet  will 
give  the  same  result,  viz. 


APPENDIX  I  399 


(5) 

This  expression  (5)  is  the  general  formula  by  which  we  may  rectify  any 
index  number  formula,  P01,  by  both  tests  at  once. 

Note  A  to  Chapter  VII,  §  19.  Crossing  the  Two  Crosses  (i.e.  the  One 
Obtained  Arithmetically  and  the  Other,  Harmonically).  While  neither  arith- 
metic nor  harmonic  crossing  of  two  time  antitheses  will  yield  an  index 
number  fulfilling  the  time  reversal  test  the  geometric  cross  of  these  two  crosses 
will  do  so  and  will  in  fact  be  identical  with  the  geometric  cross  of  the 
formulas  themselves,  as  the  reader  can  readily  prove. 

Moreover,  without  using  any  such  geometric  crossing  we  can  approach 
the  same  result  as  a  limit  by  continued  application  of  the  arithmetic  and 
harmonic  crossing  as  follows:  (1)  cross  the  original  antithetical  formulae 
arithmetically  and  harmonically ;  (2)  cross  the  last  two  results  arithmet- 
ically and  harmonically ;  (3)  again  cross  the  last  two  results  arithmetically 
and  harmonically;  and  so  on  indefinitely.  In  this  series  the  two  terms 
approach  each  other  so  rapidly  that  two  or  three  steps  will  suffice,  practi- 
cally, to  make  them  equal.  Compare  Appendix  I,  Note  to  Chapter  IX,  §  1. 

Note  B  to  Chapter  VII,  §  19.  Two  Geometric  Time  Antitheses  May  be 
Crossed  Aggregatively  as  May  Two  Aggregative  Time  Antitheses.  Any  two 
geometric  time  antitheses,  such  as  23  and  29,  may  be  written,  in  fractional 
form,  as  follows : 


23  = 

'"Vp^  x  p'0p'°«'0  x ... 

and 


x 


X 

If  written  in  the  above  form  they  may  readily  be  combined  aggregatively 
by  adding  the  two  above  numerators  for  the  new  numerator  and  adding 
the  two  denominators  for  the  new  denominator. 

Likewise  the  aggregatives  (Formulae  53  and  59)  may  be  crossed  aggre- 
gatively, the  result  being 

Spigo  + 


Each  of  these  aggregative  crosses  (the  aggregative  cross  of  the  geometries 
and  the  aggregative  cross  of  the  aggregatives)  conforms  to  the  time  test, 
as  may  readily  be  proved  by  the  twofold  procedure.  The  last  named  ag- 
gregative cross  (between  the  two  aggregative  time  antitheses)  is  interest- 
ing mathematically  because  its  factor  antithesis  turns  out  to  be  a  new 
and  curious  average  of  Formulae  53  and  59  very  different  from  any  of  the 

other  averages  used  in  this  book,  viz.  1  +  (53)  •*•  1  +  -  —  -. 

(59) 


400        THE  MAKING  OF  INDEX  NUMBERS 


These  aggregative  means  agree  closely  with  the  geometric  means. 

Thus  the  geometric  is  the  only  one  of  our  six  types  of  averages  which 
can  be  used  universally  for  crossing  formula  themselves  (any  two  time  an- 
titheses or  any  two  factor  antitheses)  so  as  to  satisfy  the  time  reversal 
and  factor  reversal  tests.  Of  the  other  types  of  average  only  the  aggre- 
gative will  satisfy  the  time  reversal  test  and  its  application  is  limited  to 
crossing  two  geometric  time  antitheses  or  two  aggregative  time  antitheses, 
as  just  shown. 

Note  to  Chapter  VIII,  §  6.  Formulce  1004,  1014,  1124,  U34,  1144  are 
Factor  Antitheses  of  1008,  1018,  1128,  1188,  1143,  Respectively,  Although 

Different  Cross-Weightings  of  53and54 
(Priced 


7J  7^  75  '16  17  'IB 

CHART  63  P.  There  is  close  agreement  between  the  four  methods  of 
crossing  weights.  The  antithesis  of  each  also  agrees  closely  with  its 
original,  being  sensibly  identical  therewith  except  in  the  last  two  cases  and 
absolutely  so  in  the  first. 

Derived  Otherwise.    We  are  to  show  that  if  (1)  P'oi  and  P"oi,  differing 
only  in  weights,  be  combined  so  as  to  form  another  formula,  P0i,  by  crossing 

their  weights,   and    if   (2)   their    factor    antitheses    [5Ml  -j-  Q'01  and 

\Spo9o 

•  Q"oi)  be  likewise  combined  to  form  another  (namely,  ?2l2>  +  QQ1]  , 
/  V  Soo         / 


the  latter  will  be  the  factor  antithesis  of  P0i. 

When  this  is  stated  algebraically  it  becomes  almost  self-evident. 

If  P'oi  and  P"oi  be  combined  into  POI,  andif  their  factor  antitheses,  namely, 


Q/OI  and 


be  combined  into 


APPENDIX  I 


401 


this  is  evidently  the  factor  antithesis  of  Poi  (Qoi  being  of  the  same  model 
as  Poi  since  by  hypothesis  the  former  is  of  the  same  model  as  P'oi  and  P"0i, 
and  the  latter  as  Q'oi  and  Q"0i,  while  all  these  four  are  of  the  same  model 
as  each  other). 

Note  to  Chapter  VIII,  §  10.     Unlike  Formula  Crossing,  Weight  Crossing 
May  be  Not  Only  Geometrically  but  Arithmetically  and  Harmonically  Done. 

Different  Cross-We/ghtings  of  55  and  54 
(Quantities) 

~~ //53*II5* 
2153.2154 
3IS4._3I53 
415* 


7J 


15  16  77 

CHART  63Q.    Analogous  to  Chart  63  P. 


1Q 


It  will  be  remembered  that  the  geometric  method  of  crossing  weights  gives 
the  same  result  from  crossing  weights  I  and  IV  as  from  crossing  weights 
II  and  777.  But  this  is  not  true  of  the  arithmetic  or  harmonic  methods  of 
crossing  weights.  Just  as  the  cross  formula,  123  and  125,  slightly  differ 
from  each  other  (as  do  133  and  135,  143  and  145),  so  do  their  cross  weight 
analogues  slightly  differ  from  each  other  if  the  crossing  is  performed 
arithmetically,  and  also  if  it  is  performed  harmonically. 

Since  crossing  the  weights  by  means  of  the  arithmetic  method  or  by 
means  of  the  harmonic  method  has  never  been  suggested  by  other  writers, 
except  as  applied  to  the  aggregative  index  number,  they  have  been  cal- 
culated here  only  for  that  type  of  index  number.  The  results  do  not,  of 
course,  differ  very  appreciably  from  those  of  the  geometric  method  and 
the  same  agreement  between  the  results  of  crossing  by  the  various  possible 
methods  would  be  found,  though  not  quite  to  the  same  degree,  if  the  other 
types  were  calculated. 

The  identification  numbers  of  the  arithmetic  cross  weighted  index  numbers 
begin  with  2000 ;  and  the  identification  numbers  of  the  harmonic  with  3000. 

As  to  those  beginning  with  4000,  Formula  4153  is  a  cross  weight  (of  53  and 
54)  by  means  of  a  weighted  arithmetic  mean  of  the  weights.  Formula  4154 
is  its  factor  antithesis  and  4353  the  cross  (geometrical)  of  4153  and  4154. 

Graphically,  Charts  63P  and  63Q  show  the  closeness  of  the  four  methods 
of  crossing  the  weights  of  Formulae  53  and  54.  They  could  scarcely  fail 


402         THE  MAKING  OF  INDEX  NUMBERS 


to  agree  closely  because  Formulae  53  and  54  are  themselves  so  close  together. 
It  is  noteworthy  that  Formula  4153  differs  more  from  its  factor  antithesis 
than  any  other  combination  of  53  and  54  differs  from  its  factor  antithesis. 

Charts  64P  and  64Q  show  the  final  result  after  double  rectification  of 
all  the  cross  weight  formulae  as  compared  with  the  cross  Formula  353. 
They  are  quite  indistinguishable  from  each  other  and  from  Formula  353. 
That  is,  all  of  the  foregoing  new  cross  weight  formulae  lie  in  practical  coin- 
cidence with  the  middle  tine  of  the  five  tine  fork.  So  close  are  the  new 
middle  tine  curves  to  Formulae  1153,  1154,  etc.,  that  the  differences  are 
of  no  practical  significance. 

It  is  worth  noting,  however,  that  of  the  four  methods  of  weight  crossing, 
namely,  those  used  in  Formulae  1153,  2153,  3153,  4153,  we  can  show  reason 
for  decided  preferences.  These  will  soon  be  discussed.  The  only  point  to  be 
emphasized  here  is  that  Formula  2153  formed  by  arithmetically  averaging 
the  weights  of  Nos.  53  and  54  is  the  only  one  of  the  four  which  necessarily 
falls  between  53  and  54,  or  necessarily  agrees  with  these  if  they  agree  with 
each  other. 

We  are  not  justified  in  taking  for  granted,  as  has  been  done  hitherto, 
that  any  cross  weight  formula  lies  between  the  two  original  formulae  (as 
is  the  case  with  cross  formulae).  Examination  shows  that  it  is  not  true  of 
the  geometric,  harmonic,  or  Formula  4153. 

Let  us  take  up  these  three  in  order.  First,  consider  the  geometric 
method  of  crossing  the  weights.  Suppose  that  of  the  price  relatives  to  be 
averaged,  half  are  100  per  cent  and  the  remaining  half  are  300  per  cent. 
Next  let  us  suppose  the  numerical  values  of  the  weights  for  the  base  year 
to  be  (for  the  first  18  relatives  of  100  each)  respectively  2,  0,  2,  0,  2,  0,  etc., 
in  alternation,  and  the  numerical  values  of  the  given  year  weights  (for  the 
same  18  relatives)  to  be  0,  2,  0,  2,  0,  2,  etc.,  in  alternation;  while  for 
the  second  18  price  relatives,  of  300  each,  the  weights  are  all  unity. 

For  convenience  we  may  tabulate : 


WEIGHTING 

Base  Year 

Given  Year 

100  per  cent 

2 

0 

100  per  cent 

0 

2 

First  half 

100  per  cent 

2 

0 

100  per  cent 

0 

2 

etc. 

300  per  cent 

1 

1 

300  per  cent 

1 

1 

Second  half 

300  per  cent 

1 

1 

300  per  cent 

1 

1 

etc. 

APPENDIX  I  403 

It  is  clear  that,  under  the  base  year  system  of  weighting,  in  the  first  half 
every  even  item  has  a  zero  weight  and  disappears  leaving  only  the  odd  terms 
to  be  averaged.  But  these  are  all  alike  (100  per  cent)  and  have  each  the 
same  weights  (2).  In  the  second  half  the  price  relatives  are  all  300  and 
have  weights  1.  It  follows  that  the  average  of  all  reduces  to  an  average  of 
nine  terms  each  weighted  as  though  it  were  two  and  18  terms  each  weighted 
once ;  in  other  words,  an  average  of  two  sets  of  18  terms  each,  or  a  simple 
average  of  100  per  cent  and  300  per  cent. 

Turning  to  the  given  year  weights  we  find  the  same  result ;  for  in  that 
case  every  odd  term  disappears  in  the  first  half,  again  leaving  nine  doubly 
weighted  100's  to  be  averaged  in  with  18  singly  weighted  300' s. 

It  follows  that  the  resulting  index  numbers  are  the  same,  whether  base 
year  weights  or  given  year  weights  are  used.  In  either  case,  we  have  the 
same  figures  300  and  100  to  be  averaged  equally  weighted,  so  that  the  aver- 
age of  300  and  100  must  be  the  same  in  both  cases.  (This  must  be  true 
whether  this  average  be  arithmetic,  geometric,  or  harmonic.  If  the  average 
is  arithmetic,  the  index  number  is  200 ;  if  geometric,  173 ;  if  harmonic,  150.) 

So  much  for  crossing  the  formula. 

When  we  cross  the  weights  the  result  is  surprisingly  different.  For  the 
weights  in  the  first  half  are  all  zero  (V2  X  0,  Vo  X  2,  V2  X  0, 
Vo  X  2,  etc.) !  The  weights  in  the  second  half  are  all  unity.  Hence, 
the  entire  first  half  disappears  and  the  average  becomes  the  average  of 
18  terms  of  300  per  cent  each,  which  is  300  per  cent. 

We  have  here,  therefore,  a  case  where  the  results  of  base  year  weighting 
and  of  given  year  weighting  agree  (being  each,  say,  200)  whereas  when  we 
take  the  geometric  mean  of  the  weights  we  get  300 ! 

It  stands  to  reason,  I  think,  that  if  base  year  weighting  and  given  year 
weighting  both  give  identical  index  numbers  (as  200),  any  mean  weighting 
which  is  worth  while  ought  to  give  the  same  result  (200),  and  not  be  ca- 
pable of  giving  a  result  (300)  larger  than  either. 

Again,  if  the  base  year  and  given  year  weighting  give  different  results, 
such  as  149  and  151,  we  may  reasonably  demand  that  the  result  of  using 
mean  weights  shall  lie  between  these  figures  instead  of  lying  far  outside, 
like  300. 

Of  course,  what  has  been  proved  by  using  zero  weights  would  be  true, 
though  in  less  degree,  if  weights  not  zero,  but  very  small,  were  used. 

This  possibility  of  miscarriage  is  even  greater  in  the  case  of  the  harmonic 
average.  For  each  harmonic  average  lies  on  the  opposite  side  of  the  geo- 
metric from  the  arithmetic. 

We  find  some  examples  of  such  miscarriages  of  the  cross  weighted 
formulae.  The  median  shows  such  a  miscarriage.  Thus  the  base  year 
weighting  (Formula  33)  gives  (for  quantities,  for  1918)  122.39  and  the 
given  year  weighting  (Formula  39),  123.50,  but  the  geometrically  cross 
weighted  median  (1133),  instead  of  lying  between  122.39  and  123.50,  is 
122.27.  A  few  of  the  chain  figures  (for  quantities  1917  and  1918)  are 
still  further  out  of  line. 

For  the  aggregative  Formula  1153  (with  geometric  cross  weights)  and 
3153  (with  harmonic  cross  weights)  the  figures  in  a  few  cases  do  not  remain 
between  those  for  53  and  54  but  likewise  jump  over  the  traces. 


404 


THE  MAKING  OF  INDEX  NUMBERS 


The  only  case  where  this  happens  with  the  geometric  cross  weights  is 
for  prices  for  1918  (chain)  where  Formulae  53  and  54  give  178.56  and  178.43 
while  1153  gives  178.37. 

The  harmonic  likewise  escapes  the  confines  of  Formulae  53  and  54  in 
several  instances  for  the  fixed  base  index  numbers.  Thus  for  prices : 

For  1917,  Formulae  53  and  54  give        162.07  and 

161.05 
whereas  Formula  3153  gives  162.11 


For  1918,  Formulae  53  and  54  give 
whereas  Formula  3153  gives 

353  Compared  with 
Its  Cross -Weight  Pivots 
(Prices) 


177.87  and 

177.43 

176.94 


|5* 


73  '14  15  '16  77  73 

CHART  64P.  On  the  score  of  accuracy  there  is  almost  no  preference 
between  the  doubly  rectified  cross  weight  formulae  and  353. 

As  to  Formula  4153,  it  presents  the  allurement  of  using  a  weighted  aver- 
age of  weights.  But  this  overdoes  the  effort  to  use  weights  somewhat  as 
a  double  negative  overdoes  negation. 

A  simple  illustration  will  suffice  to  show  that  Formula  4153  fails  to 
split  the  difference  between  53  and  54  and  that  its  results  are  unfair.  Sup- 
pose the  price  of  wheat  in  1913  was  pQ  =  $1  a  bushel  and  in  1914,  p\  =  $20 
a  bushel,  while  rubber  was  p'0  =  $20  a  pound  in  1913  and  p'\  =  $1  a 
pound  in  1914  ;  and  that  their  quantities  were  go  =  3  million  bushels  and 
g'o  =  3  million  pounds  respectively  in  1913,  and  q\  =  300  million  bushels 
and  q'i  =  300  million  pounds  respectively  in  1914.  Then,  by  Formula 
53,  we  find  the  average  price  change  of  these  two  commodities  to  be 


+  P'*'«  =  2°X3+    1X3  _  20+    1  =  cent 

Po9o  +  p'cfl'o        1X3+20X3        1+20 


APPENDIX  I 


405 


By  Formula  54,  we  have 

Piqi  +  p'iq'i  _  20  X  300  +  1  X  300  _  20  +  1 
Poqi  +  p'oq'i      1  X  300  +  20  X  300      1+20 


100  per  cent. 


Thus  Formulae  53  and  54  agree.     But  Formula  4153  does  not  lie  be- 
tween, i.e.  does  not  agree  with  both. 


Formula  4153  is 


pi  /pogo  +  pigA  +  p/i  /p'og' 

\    po  +  Pi     /  \      p' 


+ 


p'o  + 


/ 

\ 


,  i.e. 


20 


Po  +  Pi    /  \      P  o  +  Pi 

I  X  3  +  20  X  300\   ,   -  /20  X  3  +  1  X  300\ 
/         \ 


1+20 


20  +  1 


) 


lfl  X  3  +20  X300\       2Q/20  X3  +  1  X  300\ 
V  1+20          )  \          20  +  1          / 

355  Compared  with 
its  Crass-Weight  Rivals \ 


912  per  cent. 


(Quantities) 


73  74  '/$  16  77 

CHART  64Q.    Analogous  to  Chart  64 P. 


78 


Each  bracket  is  an  average.  Inside  the  brackets  the  use  of  the  prices 
1  and  20  as  weights  for  averaging  the  quantities  3  and  300  gives  the  greater 
weight  to  the  300  in  the  left  brackets  and  to  the  3  in  the  right  brackets. 
Hence  the  resulting  average,  i.e.  the  value  of  the  bracket,  is  nearer  300 
in  the  case  of  the  left  brackets  and  nearer  3  in  the  other  two.  In  other 
words,  that  quantity  always  dominates  which  pertains  to  the  year  in  which 
the  commodity  happens  to  have  the  higher  price. 

Now  it  stands  to  reason  that  this  is  unfair,  not  only  because  the  result 
(912  per  cent)  lies  outside  the  two  coincident  results  (100  per  cent)  of  For- 
mulae 53  and  54,  but  also  because  their  equality  itself  stands  to  reason. 


406         THE  MAKING  OF  INDEX  NUMBERS 

Formula  53  gives  the  index  number  when  the  quantities  are  3  and  3 ;  and 
Formula  54  gives  the  index  number  when  the  quantities  are  300  and  300. 
This  is  clearly  as  it  should  be  since  the  weighting  is  purely  relative. 
If  then  the  base  year  weighting  and  given  year  weighting  are  thus  relatively 
the  same  for  the  two  commodities  we  surely  have  no  right  to  spoil  this  same- 
ness by  any  combination  of  these  two  methods  of  weighting. 

The  numerical  example  given  shows  that  weighting  the  quantities  by 
prices  (before  averaging  them  for  use  as  weights  for  prices)  introduces  a 
wrong  principle.  While  it  does  not  bias  the  result  it  produces  a  haphazard 
favoritism,  favoring  p\  in  the  numerator  or  p0  in  the  denominator.  This 
is  unfair,  for  favoring  p\  in  the  numerator  relatively  to  p'\  in  the  numerator 
influences  the  resulting  ratio  in  the  same  direction  as  favoring  p0  in  the 
denominator  relatively  to  p'o  in  the  denominator. 

Formula  4153  represents  distinctly  the  most  erratic  of  the  methods  of 
crossing  weights.  The  geometric  will  follow  closely  the  arithmetic,  both 
being  simple ;  and  the  harmonic  will  be  close  to  the  geometric.  But  For- 
mula 4153  introduces  in  the  weighting  a  new  disturbing  element.  Accord- 
ingly, we  find  that  Formula  4153  does  not  remain  between  53  and  54  as 
often  even  as  do  1153  or  3153. 

We  find  for  prices  (fixed  base)  the  following  cases  where  Formula  4153 
falls  outside  the  range  between  53  and  54. 

For  1916  Formulae  53  and  54  give        114.08 
and  114.35 

For  1916  Formula  4153  gives  114.44 

For  1917  Formulae  53  and  54  give        162.07 

and  161.05 

For  1917  Formula  4153  gives  162.40 

For  1918  Formula*  53  and  54  give        177.87 

and  177.43 

For  1918  Formula  4153  gives  178.26 

For  quantities  we  find  similar  discrepancies  for  1918.  There  are  like  dis- 
crepancies in  the  chain  numbers. 

After  rectification  by  Test  2  the  results  (for  Formula  4353)  are  appre- 
ciably improved. 

Formula  2153  remains  as  the  only  cross  weight  formula  which  always 
and  necessarily  falls  between  53  and  54. 

Formula  2153  is  obtained  by  crossing  the  weights  of  53  and  54  arith- 
metically (by  taking  the  simple  arithmetic  average  of  their  weights).  We 
shall  show  first  that  this  cross  weight  formula  is  identical  with  the  cross 
formula  obtained  by  crossing  53  and  54  aggregatively.  In  its  role  as  a  cross 
weight  formula  (arithmetically  crossed)  it  is 


APPENDIX  I  407 


In  its  r61e  as  a  cross  formula  (aggregatively  crossed)  it  is 

+ 


That  the  two  are  identical  is  evident  by  canceling  the  "2's"  in  the  first 
and  multiplying  out. 

The  last  formula,  being  a  mean  or  average  of  53  and  54,  must  necessarily 
lie  between  53  and  54,  which  was  to  have  been  proved. 

In  this  connection  it  is  interesting  to  note  that,  besides  Formula  2153, 
there  could  be  constructed  other  formulae  which  are  both  cross  formula 
and  cross  weight  formulae.  Formula  2153  is  such  as  between  53  and  54,  aggre- 
gative index  numbers.  But  similar  results  can  be  had  with  arithmetic 
index  numbers  and  also  with  harmonic  index  numbers.  In  each  of  these 
cases  we  get  precisely  the  same  result  by  taking  two  formulae  (say,  3  and  9, 
or  5  and  7,  or  13  and  19,  or  15  and  17)  of  the  same  model  and  crossing  their 
weights  arithmetically  as  by  crossing  the  formulce  themselves  aggregatively. 

Note  to  Chapter  IX,  §  1.  The  (Geometric)  Cross  of  Formulce  8053  and 
8054  is  Identical  with  353.  Using  a  for  Formula  53,  and  b  for  54,  8053  is 

,  8054  is  -  .    Their  cross  or  geometric  mean  is 


Vab  =  V53  X  54  =  353. 


Note  to  Chapter  XI,  §  4.  //  the  Mode  is  Above  the  Geometric  Forward  It 
is  Below  Backward.  This  is  most  easily  made  evident  by  considering 
Charts  IIP  and  11Q.  We  saw  that  the  arithmetic  forward  and  backward 
are  not  prolongations  of  each  other  because  the  arithmetic  fails  to  satisfy 
Test  1 ;  and  the  same  is  true  of  the  harmonic  forward  and  backward.  But 
for  any  formula  which  does  satisfy  Test  1,  the  forward  and  backward  forms 
will  be  prolongations  of  each  other.  This  is  true  of  all  the  simple  index 
numbers  (except  the  arithmetic  and  harmonic)  including  the  geometric 
and  mode.  Consequently,  we  have  the  picture  simply  of  two  straight 
lines  intersecting  at  the  origin,  one  for  the  geometric  forward  and  back- 
ward, and  the  other  for  the  mode  forward  and  backward.  It  is,  therefore, 
clear  that  if  on  one  side  of  the  origin  the  mode  lies  above  the  geometric, 
it  must  lie  below  it  on  the  other. 

Note  to  Chapter  XI,  §  10.  Derivation  of  Probable  Error  of  Any  of  the  13 
Formulce  Considered  as  Equally  Good  Observations.  Assuming  that  the  13 
index  numbers  are  equally  good,  the  formula  for  their  probable  error,  i.e. 
the  as-likely-as-not  deviation  (from  their  mean)  of  any  of  the  13  observa- 
tions taken  at  random  is  .6745\/ where  d  denotes  the  deviation  from 

\n  —  1 

their  mean  of  any  of  the  observations,  and  n  denotes  the  number  (in  this 
case  13)  of  the  observations. 


408         THE  MAKING  OF  INDEX  NUMBERS 

The  expression  for  the  "probable  error"  of  the  mean  itself  is  the  preced- 
ing expression  divided  by  Vn. 

Note  to  Chapter  XI,  §11.  Does  "Skewness"  of  Dispersion  Matter? 
Hitherto  one  of  the  chief  questions  investigated  by  students  of  index  num- 
bers is  the  question  of  the  distribution  of  the  data  averaged,  the  sort  of 
dispersion,  whether  in  particular  it  is,  or  is  not,  "skew."  Thus  we  know, 
from  the  work  of  Wesley  C.  Mitchell  and  others,  that  price  relatives  dis- 
perse far  more  widely  upward  than  downward,  the  reason  obviously  being 
that  there  is  more  room  for  dispersion  upward.  In  the  downward  direc- 
tion they  are  limited  by  zero  while  upward  there  is  no  limit. 

It  has  been  assumed  that  the  character  of  this  distribution  will  have  a 
determining  influence  in  the  choice  of  the  best  index  number.  Much  is 
made  of  this  consideration  by  Walsh,  Edgeworth,  and  others.  Elaborate 
arguments  have  been  constructed  to  show  that  the  geometric  mean  or  some 
other  is  the  appropriate  mean  to  use  in  constructing  index  numbers  based 
on  the  idea  that  the  dispersion  is  supposed  to  be  more  symmetrical  "geo- 
metrically" than  it  is  "arithmetically." 

It  will  be  noted  that  in  this  book  we  have  had  no  occasion  whatever  to 
invoke  this  consideration.  In  choosing  the  formula  for  an  index  number 
the  skewness  or  asymmetry  of  distribution  of  the  terms  averaged  is 
of  absolutely  no  consequence.  This  may  seem  a  most  revolutionary 
idea.  There  has  been  a  growing  tendency  to  take  account  of  the 
distribution  of  the  data  in  any  social  problem  before  deciding  on 
whether  the  geometric  or  the  arithmetic  process  of  averaging  should  be 
used.  I  am  offering  no  objection  to  this  in  general.  On  the  contrary  it 
is  of  great  importance  for  many  purposes  in  social  problems.  Even  aver- 
aging human  heights  and  weights  should  take  the  character  of  the  distri- 
bution into  account. 

But  in  the  realm  of  index  numbers  the  case  is  different  and  for  a  very 
simple  reason.  Unlike  heights  or  weights,  price  relatives  or  quantity  rela- 
tives are  ratios  of  two  terms  either  of  which  two  may  be  taken  as  the  nu- 
merator. Any  ratio  is  necessarily  a  double  ended  affair.  If  used  in  one  di- 
rection the  ratios  disperse  in  one  way  while  if  used  in  the  other  direction 
they  disperse  in  precisely  the  opposite  way.  The  large  ratios  for  one  of 
the  two  ways  become  the  small  ratios  the  other  way  and  in  the  same  rela- 
tive degree.  Thus,  if  sugar  rises  from  10  cents  to  20  cents  and  wheat  from 
$1  to  $3  between  two  times  or  places  the  price  relatives  are  200  per  cent 
and  300  per  cent,  the  wheat  relative  being  a  half  greater  than  the  other. 
But,  reversing  the  direction  of  the  comparison,  the  price  relatives  are  50  per 
cent  and  33i  per  cent,  the  sugar  relative  being  now  a  half  greater  than  the  other. 

Charts  IIP  and  IIQ  illustrate  the  reversal  of  the  dispersion  through  the 
reversal  of  the  times. 

When,  therefore,  we  rectify  by  Test  1  thus  taking  account,  in  equal  terms, 
of  these  two  opposite  dispersions,  any  skewness  of  distribution  enters  in 
both  ways  and  cancels  itself  out.  Consequently,  in  our  final  results,  such 
as  309,  323,  and  353,  there  is  no  trace  of  any  effect  of  skewness.  These 
three,  so  far  as  they  differ  at  all,  differ  sometimes  in  one  direction  and 
sometimes  in  the  other,  although  309,  for  instance,  is  made  up  from  index 
numbers  affected  greatly  by  skewness  of  distribution. 


APPENDIX  I 


409 


Distribution  of  1457  Price  Relatives 
(Forward  and  Backward) 


1000-  , 


i 
I 


100- 


10- 


BACKWARD 


DUMBER   OF  COMMODITIES 

CHART  65.  Showing  how,  in  the  ratio  chart, 
the  distribution  of  the  price  relatives,  taken  for- 
ward and  backward,  is  exactly  reversed  in  skew- 
ness  and  order  of  averages  except  as  to  the  arith- 
metic and  harmonic  (which  exchange  places). 


410         THE  MAKING  OF  INDEX  NUMBERS 

If  we  plot  the  two  distributions  on  an  ordinary  frequency  curve  such  as 
Chart  62  it  is  true  that  the  dispersion  in  both  cases  will  be  wider  at  the 
top  than  at  the  bottom  (or,  as  it  is  usually  plotted,  at  the  right  than  at  the 
left).  But,  and  this  is  significant,  the  commodities  which  are  at  the  top 
in  one  case  are  at  the  bottom  in  the  other  and  vice  versa. 

The  real  reason  for  the  greater  dispersion  upward  than  downward  lies 
in  the  arithmetical  scale  by  which  we  measure.  If  we  use  the  ratio  chart 
we  cannot  even  say  that  the  distribution  is  skew,  and  if  skew,  in  any  par- 
ticular direction.  Chart  65  shows  the  distribution  of  the  1437  price  rela- 
tives of  the  War  Industries  Board  for  1917  relatively  to  the  year  July,  1913- 
July,  1914  and  (in  fainter  and  dotted  lines)  the  distribution  reversed.  It 
will  be  observed  that  the  skewness  is  reversed,  the  mode  being  the  least 
of  the  five  averages  in  the  original  distribution  and  the  greatest  in  the 
dotted  figure.  The  order  of  the  five  averages  is  reversed  in  the  two  dis- 
tributions except  that  the  arithmetic  and  harmonic  exchange  places  as 
usual.  When  ratio  charting  is  used  we  may  say  that  a  "normal  distri- 
bution" is  one  which  is  symmetrical  about  the  mode,  or  geometric,  or  me- 
dian, which  three  will  normally  agree,  while  the  arithmetic  will  always  be 
above  and  the  harmonic  below  these  three ;  there  is  exactly  as  much  chance 
of  skewness  in  one  direction  as  the  other. 

Note  to  Chapter  XI,  §  13.  Formula  53  and  54  are  Sometimes  Slightly 
Biased.  Whether  54  is  greater  or  less  than  53  depends  on  whether  the 
price  relatives  are  positively  or  negatively  correlated  with  the  quantity 
relatives. 

The  price  relatives  and  the  quantity  relatives  (1913  being  the  base)  for 
the  36  commodities  used  here  are  correlated  as  follows : 

1914  +  .265 

1915  +  .023 

1916  +  .035 

1917  -  .133 

1918  -  .250 

These  correlations  are  mostly  too  small  to  have  much  significance  and 
are  about  equally  positive  and  negative.  A  clearer  and  more  consistent 
case  of  correlation  between  price  and  quantity  movements  is  given  by  Pro- 
fessor Persons,  who  finds  that  for  12  leading  crops  the  price  and  quantity 
movements  are  negatively  correlated  with  the  high  coefficient  of  —  .88. 
When  the  correlation  is  positive  it  means  that  the  weights  (i.e.  the  g's) 
in  Formula  53,  which  has  the  system  called  weighting  7,  are  analogous  in 
this  respect  to  the  system  weighted  /  for  all  the  other  types  of  index  num- 
bers. It  will  be  remembered  that  for  the  arithmetic,  harmonic,  geometric, 
and  median  (and,  theoretically,  the  mode),  weighting  /  imparted  a  down- 
ward bias  and  weighting  IV  imparted  an  upward  bias. 

This  was  due  to  the  price  element  in  the  weights  which  in  weighting  IV 
tended  to  associate  a  large  weight  with  a  large  price  relative  and  a  small 
weight  with  a  small  price  relative,  thus  overweighting  the  high  and  pro- 
ducing the  upward  bias;  with  weighting  7  the  opposite  situation  holds 
true. 

But  in  the  case  of  the  aggregative  type,  the  weights  contain  no  price 


APPENDIX  I  411 

element,  as  the  weights  are  mere  quantities.  Yet  the  same  effect  is  pro- 
duced if  these  quantities  are  positively  correlated  with  the  price  move- 
ments ;  for  we  then  have  the  same  tendency  to  an  association  of  large 
weights  with  the  large  price  relatives  and  small  with  the  small ;  only  that 
tendency  is  much  weaker  —  unless  the  correlation  is  100  per  cent  so  that 
the  quantities  behave  exactly  as  though  they  were  prices. 

We  would  expect,  then,  that  wherever  correlation  is  positive  we  would 
find  the  aggregative  IV,  or  59  (or  54),  above  the  aggregative  /,  or  53,  just 
as  we  found,  for  the  other  types,  3  below  9,  13  below  19,  23  below  29,  and 
33  below  39.  And  this  is  just  what  we  do  find  except  that  the  differences 
for  the  aggregative  are  much  less  than  for  the  other  types.  On  the  other 
hand,  where  the  correlation  is  negative  we  would  expect  to  find  the  oppo- 
site and  so  we  do.  That  is,  in  our  calculations  for  the  36  commodities,  53 
lies  below  54  (or  59)  in  1914,  1915,  1916,  when  the  correlation  is  positive, 
but  above  in  1917  and  1918  when  the  correlation  is  negative.  This  is 
shown  in  the  upper  tier  of  Charts  39P  and  39Q. 

But  for  Persons'  figures  for  prices  and  quantities  of  12  crops  (relatively 
to  the  base  1910)  53  is  always  (except  once)  above  54  (or  59)  showing  a 
definite  upward  bias  of  53  due  to  the  definite  and  high  negative  correlation, 
i.e.  to  the  fact  that  big  crops  make  for  low  prices  and  vice  versa.  This  is 
shown  in  Charts  47P,  47Q,  48P,  48Q. 

Some  readers  will  be  asking  whether  there  is  not  always  some  upward 
bias  in  53  and  downward  bias  in  54  aside  from  mere  error  in  either  direction. 
The  answer  is  that,  while  in  the  case  of  crops  a  negative  correlation  is  found 
because  crops  here  represent  supply,  prices  are  affected  also  by  demand, 
and  the  quantities  in  our  formulas  are  about  as  likely  to  represent  changes 
in  demand  as  changes  in  supply.  As  prices  go  up  with  increased  demand 
and  down  with  increased  supply  the  chances  seem  about  even  whether  the 
actual  quantities  marketed  will  be  positively  or  negatively  correlated  with 
prices,  and  all  the  figures  we  have,  except  these  crop  figures,  sustain  this 
conclusion. 

Moreover,  the  same  logic  applies,  not  only  to  this  comparison  of  aggre- 
gatives,  but  to  comparisons  of  two  arithmetics,  or  two  harmonics,  etc., 
where  the  weighting  systems  differ  only  as  to  the  quantity  element.  In 
all  these  cases  weighting  /  and  weighting  II  differ  from  each  other  only  as 
to  the  quantities  as  do  III  and  IV  from  each  other.  Thus  the  same  rea- 
soning by  which  aggregatives  7  and  IV  differ  applies  to  arithmetic  I  and  77, 
or  to  arithmetic  777  and  IV  as  well  as  to  the  corresponding  harmonics  and 
the  corresponding  geometries.  An  inspection  of  the  charts  shows  just  what 
we  are  thus  led  to  expect.  In  all  these  cases,  both  for  the  price  indexes 
and  the  quantity  indexes,  with  trifling  exceptions,  the  7  is  below  the  77 
(and  the  777  below  the  7F)  for  1914,  1915,  and  1916  and  above  for  1917 
and  1918. 

The  only  exceptions  are  for  the  quantity  indexes  of  1918  for  the  har- 
monics and  geometries  where  77  very  slightly  exceeds  7,  presumably  owing 
to  some  disturbing  influence  of  the  greatly  aberrant  quantity,  skins. 

This  faithful  correspondence  between  correlation  coefficients  and  the 
influence  of  quantities  in  the  weighting  of  the  price  index  is  certainly  re- 
markable when  we  consider  how  infinitesimal  are  the  influences  thus  traced. 


412         THE  MAKING  OF  INDEX  NUMBERS 

Even  the  sluggish  median  reflects  the  same  influences  with  few  exceptions. 
We  can  also  say  that  almost  always  the  larger  the  correlation  coefficient 
the  larger  the  divergence  found  between  53  and  54.  Thus  the  behavior 
of  all  of  our  weighting  systems  has  been  pretty  fully  analyzed.  The 
large  differences  made  (to  price  indexes)  are  those  made  by  price  elements 
in  the  weights  and  the  small  by  the  quantity  elements  in  proportion  to 
their  correlation  with  the  price  relatives. 

We  see,  then,  that  Laspeyres'  and  Paasche's  formulae  (53  and  54)  are 
usually  close  to  each  other  even  when  slightly  biased.  In  order  to  study 
the  consequences  of  a  really  wide  difference  between  them  we  pick  out 
from  among  our  36  commodities  "rubber"  and  "skins"  and  calculate  the 
index  number  for  these  two  only,  and  then  do  the  same  for  "lumber"  and 
"wool."  The  first  pair  are  chosen  to  make  53  most  exceed  54,  and  the 
second  to  make  54  most  exceed  53.  The  reason  is  that  the  first  pair, 
rubber  and  skins,  happen,  during  the  period  covered,  to  have  had  their 
prices  most  affected  by  supply  so  that  their  quantities  and  prices  tended  to 
move  in  opposite  directions.  The  quantity  of  rubber  marketed  rose  and 
its  price  fell;  the  quantity  of  skins  fell  and  the  price  rose  enormously. 
Lumber  and  wool,  on  the  other  hand,  were  affected  chiefly  by  demand.  An 
increase  of  demand  drove  up  the  price  of  wool  much  beyond  the  average 
rise  of  prices,  while  the  quantity  marketed  also  increased ;  contrariwise,  a 
decrease  of  demand  kept  the  price  of  lumber  far  behind  the  average  while 
the  quantity  marketed  decreased. 

That  is,  the  p's  and  g's  of  rubber  are  correlated  negatively,  as  are  those  of 
skins,  while  the  p's  and  g'sof  lumber  are  correlated  positively  as  are  those  of  wool. 

As  we  have  seen,  when  negative  correlation  prevails,  53  exceeds  54,  and 
when  positive,  54  exceeds  53.  In  the  present  case  the  figures  are  as  given 
in  Table  52  and  Table  53.  Here,  occasionally,  are  considerable  differences 
between  the  results  obtained  by  using  Formula  53  or  Formula  54.  In 
the  less  extreme  case  of  lumber  and  wool,  the  maximum  excess  of  54  over 
53  is  only  about  eight  per  cent  (for  1918),  while,  in  the  much  more  extreme 
case  of  rubber  and  skins  53  exceeds  54  by  32  per  cent  in  1918. 

One  reason  why  the  figures  were  worked  out  for  such  non-representative 
cases  was  to  discover  whether  Formula  2153  would  still  be  able  to  serve  as  a 
good  short  cut  for  353.  Table  54  and  Table  55  show  that  it  would  be  a  good 
substitute  for  the  less  extreme  case  of  lumber  and  wool,  but  not  always 
very  good  for  the  other. 

It  will  be  seen  that,  in  the  less  extreme  case  of  lumber  and  wool  2153  devi- 
ates from  353  more  than  a  third  of  one  per  cent  in  only  one  instance,  that  of 
quantities  in  1918,  when  the  deviation  amounts  to  nine  tenths  of  one  per 
cent.  In  the  more  extreme  case  of  rubber  and  skins,  2153  deviates  by 
over  one  per  cent  four  times  out  of  ten,  the  deviation  reaching  five  per 
cent  for  prices  in  1918  (when  53  exceeds  54  by  32  per  cent  and  353Q  is  over 
200  per  cent).1  Such  deviations  are,  of  course,  quite  impossible  when,  in- 
stead of  two  culled  commodities,  a  larger  number  of  commodities,  unculled, 
are  included. 

Note  to  Chapter  XII,  §  1.  Method  Used  for  Ranking  Formula  in  Close- 
ness to  853.  The  method  of  ranking  the  134  index  numbers  relatively  to 
» See  Appendix  I,  Note  to  Chapter  XV,  §  2. 


APPENDIX  I  413 

Formula  353  as  ideal  consists  in :  (1)  finding  the  difference  between  any 
given  index  number  and  the  ideal  for  each  year  (1914-1918) ;  (2)  reducing 
these  differences  to  percentages  of  the  ideal  index  number;  (3)  further 
adjusting  them  in  inverse  proportion  to  the  dispersion  index  referred  to  in 
Appendix  I,  Note  to  Chapter  V,  §  11;  and  (4)  taking  the  simple  arith- 
metic average  of  these  deviations  disregarding  plus  and  minus  signs. 

This  method  of  grading  our  formulae  is  not  the  most  accurate  possible 
but  is  accurate  enough  for  our  purpose  and  much  more  easily  computed 
than  the  most  accurate.  The  resulting  order  of  formulae  is  probably  al- 
most exactly  the  same  as  if  a  greater  refinement  of  method  were  employed. 
The  third  step  is  inserted  on  the  theory  that  a  year  of  very  wide  dispersion, 
like  1917,  would  naturally  show  wider  differences  among  formulae  than 
would  a  year  of  small  dispersion,  like  1914,  and  that,  therefore,  in  reckoning 
the  distance  of  any  index  number  from  the  ideal  a  small  percentage  distance 
in  1914  should  count  as  much  as  a  large  one  in  1917. 

Note  to  Chapter  XIII,  §  1.  The  Algebraic  Expression  of  the  Circular 
Test.  Let  the  three  cities,  or  years,  be  designated  as  1,  2,  and  3,  and  let 
the  index  numbers  representing  the  ratios  between  their  price  levels  be 
PIZ,  Pas,  PSI  (and  also,  of  course,  their  reverse,  P2i,  P82,  PIS).  The  pro- 
posed test  is  that  any  particular  index  formula  should  yield  results  which 
will  make  Pa2  X  P28  =  PIS  or  will  make  Pi2  X  P28  X  Psi  =  1.  These 

two  conditions  are  equivalent  if  Pi8  =  —  (i.e.  if  our  "time  reversal"  test 

PSI 

is  satisfied)  as  is  evident  by  substituting  — -  for  PIS  in  the  first  formula 

PSI 

(Pi2P28  =  PIS),  and  clearing  fractions.  The  result  is  evidently  the  second 
(P^PusPsi  =  1).  In  other  words,  the  product  of  the  three  index  numbers 
taken  in  the  same  direction  around  the  triangle  is  required,  by  the  supposed 
test,  to  be  unity. 

Note  A  to  Chapter  XIII,  §  4.  The  Simple  or  Constant  Weighted  Geo- 
metric (9021}  Conforms  to  the  Circular  Test.  That  the  simple  geometric 
(21)  or  constant  weighted  geometric  (9021)  conforms  to  the  circular  test 
is  easily  shown.  Formula  9021  is 


s)-x(£')  x... 

V          \P  o/ 

where  to,  to',  etc.,  are  constant  weights,  i.e.  the  same  for  all  the  years,  0,  1, 
etc.  The  above  formula  is  written  for  the  index  number  of  year  "  1 "  rela- 
tively to  year  "0,"  i.e.,  as  we  pass  from  "0"  to  "1."  Passing  from  "1" 
to  "2"  we  have  the  following : 


and  to  complete  the  circuit,  passing  from  "2"  to  "0," 


P 


414         THE  MAKING  OF  INDEX  NUMBERS 


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APPENDIX  I 


415 


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416         THE  MAKING  OF  INDEX  NUMBERS 

Multiplying  all  three  together  we  have 


Po      Pi       Pz/  \p'o 

which  by  cancellation  reduces  to  unity,  thus  satisfying  the  circular  test. 
The  above  proof  includes  the  simple  geometric  as  a  special  case  simply  by 
putting  w  =  wf  =  w"  =  .  .  .  =  1. 

That  the  simple  aggregative  (51)  or  constant  weighted  aggregative  (9051) 
conforms  to  the  circular  test  is  likewise  easily  shown.  Formula  9051  is 

for  the  step  from  0  to  1 
for  the  step  from  1  to  2 

for  the  step  from  2  to  0 

the  product  of  which  is  unity.  This  includes  the  simple  as  a  special  case 
where  w  =  w'  =  w"  =  .  .  .  =  1. 

Note  B  to  Chapter  XIII,  §  4.  A  Formula  Fulfilling  Tests  1  and  2  May  be 
Modified  to  Fulfill  the  Circular  Test  as  Applied  to  Three  Specific  Dates.  It 
may  interest  the  mathematically  inclined  reader  to  observe  that,  not  only 
can  conformity  to  the  circular  test  be  gained  by  making  the  weights  arti- 
ficially constant  in  the  face  of  the  facts,  but  that  such  conformity  limited 
to  a  specific  triangle  of  dates  can  be  attained  by  a  mutual  adjustment  of 
the  true  formulae  for  the  three  inconsistent  comparisons. 

Let  the  original  index  number  be  POI  which  may  be  of  any  variety.  Let 
its  rectification  by  Test  2,  in  the  usual  way,  be  P'oi.  Let  the  time  antith- 
esis of  P'oi  be  1/P'io.  With  these  P"s  as  our  starting  point  we  are  to  de- 
rive P'"s  which  fulfill  Test  1  and  from  these  P'"s  we  are  to  derive  P""s 
which  fulfill  the  circular  test  (so  far  as  the  three  particular  dates  are  con- 
cerned). 

The  rectification  by  Test  1  of  P"oi,  is  evidently,  by  the  usual  method, 

**«  =  \l — -,  which,  by  multiplying  numerator  and  denominator  by 

'P   10 

VP'01  reduces  to    , 
Likewise  P"10  -       P/1° 


VP'10P 


01 


the  last  two  expressions  having  the  same  denominators. 

It  is  easy  to  show  (by  multiplying  the  last  two  equations  together),  and 
in  fact  it  has  previously  been  shown,  that  these  rectified  formulae  fulfill 
the  time  reversal  test,  i.e.  that 

••  1. 


APPENDIX  I  417 

This  may,  for  present  convenience,  be  called  the  circular  test  applied  to 
two  dates.  (Unlike  what  follows  for  three  dates,  this  two  date  test  applies 
to  any  two  dates.) 

Next,  for  the  three  specific  dates  or  places,  1,  2,  3,  such  as,  say,  Georgia, 
Norway,  and  Egypt,  we  are  to  secure  further  "rectified"  P""s  such  that 

P'"l2   X   P'"23   X   P'"31    =    1. 

These  required  formulae  are 


P"'«  = 

For  proof,  multiplying  the  above  three  equations  together,  we  have 

D///      D///      *DHt 12          28          31  _    II 


which  was  to  have  been  proved. 

Moreover,  in  obtaining  the  P"  formulae  which  satisfy  the  circular  test, 
we  have  not  lost  the  fulfillment  of  the  time  reversal  test  for  two  dates,  nor 
lost  the  fulfillment  of  Test  2.  This  applies  to  the  P""s  as  well  as  the  P'"s. 
For  instance,  as  to  Test  1, 

T>f 

P'"         = 

and,  multiplying  this  by  the  first  formula  above,  we  have 

P"i2  X  P"n 


(P"23  x  P"82)  (P"81  x  P"i3) 
1 


=  1 


•\/i  x  i  x  i 

which  was  to  have  been  proved. 

If  desired,  by  retracing  our  steps,  by  successive  substitutions,  we  can,  of 
course,  obtain  P"fn  in  terms  of  Pi2,  etc. 

Thus  the  P'",  formulae  satisfy  the  circular  test  both  as  applied  to  the 
three  particular  dates,  and  as  applied  to  any  two  dates  (time  reversal  test). 

But  this  fulfillment  of  the  circular  test  applies  only  to  three  specific 
dates.  If  we  change  date  3  this  will  change  P"'ia.  The  index  number  be- 
tween dates  1  and  2  has  thus  no  fixed  value  but  has  a  different  value  for 
every  different  date  3.  Moreover,  if  we  attempt  to  go  further,  and  find 
a  formula,  P""  which  satisfies  the  circular  test  for  four  dates,  such  that  it 
will  still  hold  for  every  three  and  for  every  two,  we  encounter  difficulties; 
for  the  P""i2  which  fulfills  the  circular  test  for  1,  2, 3,  will  differ  from  that 
which  fulfills  the  circular  test  for  1, 2, 4.  We  shall  not  even  have  a  single 
value  of  P""i2  which  can  serve  in  all  comparisons,  dual,  triple,  quadruple. 


418         THE  MAKING  OF  INDEX  NUMBERS 

Note  to  Chapter  XIII,  §  5.  The  Meaning  of  "Equal  and  Opposite  "  Cir- 
cular Gaps. 

Let  Pi2P23  .  .  .  Pm  =  1  +  a 
Let  Q,2Q28  .  .  .  Qm  =  1  -  &, 

where  a  and  b  represent  the  circular  "gaps."  Since  we  are  assuming 
that  Test  2  is  fulfilled,  Pi2  X  Qi2  =  1 ;  P23  X  Q23  =  1 ;  .  .  . ;  Pm&i  =  1 ; 
and,  therefore, 

(Pl«Q»)    (P23Q23)    •    •    •    (PnlQnl)    =   1, 

i.e. 

(1  +  o)  (1  -  6)  -  1, 

which  is  the  theorem  which  was  meant  when  it  was  stated,  for  brevity,  that 
the  deviations  a  and  b  were  "equal  and  opposite." 

That  is,  1  +  a  and  1  —  6  are  reciprocals.  Moreover,  if,  as  in  the  case 
of  Formula  353,  a  and  6  are  very  small  they  will  also  be  numerically 
equal,  to  several  decimal  places. 

Note  to  Chapter  XIII,  §  8.  For  Formula  Failing  in  Test  1  It  Makes  a 
Difference  Whether  or  Not  We  Pass  All  Around  the  Triangle  in  One  Direction. 
In  case  the  index  number  under  consideration  does  not  obey  the  time  re- 
versal test,  the  dark  vertical  line  does  not,  strictly  speaking,  measure  the 
deviations  from  the  circular  test,  if  by  that  phrase  is  meant  the  discrepancy 
found  after  going  all  around  the  triangle  in  one  direction.  In  such  a  case 
the  dark  vertical  line  for  1915  is  the  discrepancy  found  by  going  from  1913 
around  two  sides  of  the  triangle  in  one  direction  (e.g.  from  1913  to  1914  and 
then  to  1915)  and  comparing  the  position  thus  reached  with  that  reached 
by  another  start  from  1913  in  the  opposite  direction  along  the  third  side, 
i.e.  from  1913  to  1915.  In  such  cases,  where  the  time  reversal  test  is  not 
fulfilled,  there  are  thus  several  discrepancies  pertaining  to  any  triangle  of 
comparisons  (instead  of  only  one  as  for  Formula  353  and  the  other  formulae 
which  do  fulfill  this  test).  Taking  the  years  1,  2,  3,  we  have  the  circular 
gap,  1-2-3-1  or  3-2-1-3 ;  also  the  following  others :  1-2-3  compared  with 
1-3  ;  2-3-1  compared  with  2-1 ;  3-1-2  compared  with  3-2 ;  3-2-1  com- 
pared with  3-1.  But,  in  the  case  where  the  time  reversal  test  is  fulfilled,  all 
these  deviations  reduce  to  the  same,  except  that  the  reversing  of  the  direc- 
tion around  the  triangle  has  the  effect  of  changing  the  sign  of  the  figure, 
so  to  speak.  Thus  the  triangular  ratio  for  0-1-2  is  100.35  per  cent  in 
Table  34  while,  for  2-1-0,  it  is  1/100.35  per  cent,  or  99.65  per  cent,  so 
that,  in  the  first  case,  the  triangular  deviation  from  unity  is  +.35  per  cent 
and,  in  the  second,  —  .35  per  cent. 

Note  to  Chapter  XIII,  §  9.  The  Relation  of  This  Book  to  the  Appendix  on 
Index  Numbers  in  the  Author's  "  Purchasing  Power  of  Money."  This  book 
has  centered  on  the  idea  of  reversibility  as  the  supreme  sort  of  test  for  an 
index  number.  In  my  earlier  book,  The  Purchasing  Power  of  Money,  in 
the  Appendix  to  Chapter  X,  I  have  employed  other  tests.  The  difference 
between  the  two  studies  is  one  of  emphasis.  Nothing  in  the  earlier  study 
needs  to  be  abandoned  (with  the  exception  of  the  circular  test),  and  the 
conclusions  of  that  study  are,  in  general,  consistent  with  those  of  the  present 


APPENDIX  I 


419 


study.  The  fundamental  difference  in  method  between  the  two  is  that, 
in  the  earlier  study,  attention  was  concentrated  on  the  algebraic  properties 
of  the  formulae  whereas,  in  the  present,  attention  is  concentrated  on  their 
numerical  results. 

The  present  study  had  its  origin  in  the  attempt  to  compare  the  numerical 
results  of  formulae  differing  in  algebraic  properties.  But  as  soon  as  these 
numerical  results  were  calculated,  they  revealed  new  directions  in  which 
to  study  the  reasons  for  the  differences  and  similarities,  directions  of  study 
of  far  greater  practical  importance  than  the  algebraic  properties  of  the 
formulae. 

But  now  that  our  new  study  is  completed,  we  may  compare  it  with  the 
old.  In  the  old,  44  formulae  were  studied,  the  original  numbering  of  which, 
translated  into  our  new  numbering,  is  as  follows : 


TABLE  56.  CROSS  REFERENCES  BETWEEN  THE  NUMBERS 
FOR  FORMULAE  TABULATED  IN  "THE  PURCHASING 
POWER  OF  MONEY"  AND  THE  NUMBERING  USED 
IN  THIS  BOOK 


NUMBER  IN  "  PUR- 
CHASING POWER 
OF  MONEY" 

NEW 
NUMBER 

NUMBER  IN  "  PUR- 
CHASING POWER 
OF  MONEY" 

NEW 
NUMBER 

1 

51 

23 

4153 

2 

52 

24 

4154 

3 

1 

25 

9 

4 

2 

26 

10 

5 

11 

27 

7 

6 

12 

28 

8 

7 

21 

29 

9001 

8 

22 

30 

omitted 

9 

31 

31 

15 

10 

32 

32 

16 

11 

54 

33 

13 

12 

53 

34 

14 

13 

8053 

35 

29 

14 

8054 

36 

30 

15 

353 

37 

23 

16 

353 

38 

24 

17 

2153 

39 

27 

18 

2154 

40 

28 

19 

3153 

41 

25 

20 

3154 

42 

26 

21 

1153 

43 

omitted  * 

22 

1154 

44 

omitted 

*  But  calculated  in  Appendix  III. 


420         THE  MAKING  OF  INDEX  NUMBERS 

There  were,  in  the  earlier  study,  eight  tests,  each  of  which  was  applied 
in  two  ways,  first,  for  dual  comparisons  (between  two  years  only)  and, 
secondly,  for  comparing  a  series  of  years.  In  the  folding  table  opposite 
p.  418  of  the  Purchasing  Power  of  Money  each  index  number  was  credited 
with  a  score  of  "£"  for  every  test  which  it  fulfilled  in  a  dual  comparison 
only  and  "1"  for  every  test  which  it  fulfilled  in  a  series  of  years.  Since, 
as  we  have  seen,  in  Chapter  XIII  of  this  book,  only  dual  comparisons  have 
theoretical  validity,  we  here  ignore  the  distinction  between  the  "i"  and  "  1." 
In  the  earlier  study  each  test  was  stated  with  reference  to  the  applica- 
tion of  the  formula  to  the  equation  of  exchange  to  fulfill  which  any 
formula  for  prices  must  be  accompanied  by  its  factor  antithesis  (there 
called  simply  its  "antithesis")  for  the  quantities.  Each  test  was  stated 
both  in  reference  to  prices  and  quantities,  and  the  fulfillment  of  either  was 
credited  as  a  good  mark  for  the  other,  its  factor  antithesis,  because  the 
two  were  running  mates  in  the  equation  of  exchange.  Inasmuch  as  we 
here  seek  to  rectify  the  formulae  so  that  the  running  mates  may  be  of  the 
same  kind,  there  is  no  real  need  of  such  mutual  crediting.  We  need  con- 
sider, therefore,  only  the  tests  for  one  of  the  two  factors,  say,  for  prices 
(p's)  and  omit  those  separately  stated  (Tests  2,  4,  6)  for  quantities  (q's). 
We  may  also  ignore  Test  7,  "changing  of  the  base,"  as  this  has  been 
fully  considered  in  the  present  book. 

There  are  left  four  tests  included  in  the  old  book  and  not  hitherto  made 
use  of  in  the  new,  namely :  (1)  Proportionality.  An  index  number  of  prices 
should  agree  with  the  price  relatives  if  those  agree  with  each  other.  (2)  De- 
terminateness.  An  index  number  of  prices  should  not  be  rendered  zero, 
infinity,  or  indeterminate  by  an  individual  price  becoming  zero.  (3)  With- 
drawal or  Entry.  An  index  number  of  prices  should  be  unaffected  by  the 
withdrawal  or  entry  of  a  price  relative  agreeing  with  the  index  number. 
(4)  Commensurability.  An  index  number  of  prices  should  be  unaffected  by 
changing  any  unit  of  measurement  of  prices  or  quantities. 

The  last  test  eliminates  all  of  the  "ratios  of  averages"  as  shown  in  Ap- 
pendix III  and  also  Formula  51  in  our  numbered  series,  together  with 
those  derived  from,  or  dependent  on  51,  viz.  52  and  251.  All  the  other 
formulae  obey  this  test,  which  may  be  considered  of  fundamental  impor- 
tance in  the  theory  of  index  numbers. 

The  test  of  proportionality  is  really  a  definition  of  an  average.1  It  is 
fulfilled  among  the  primary  formulae  by  all  the  odd  numbered  formulae. 
But  none  of  the  even  numbered  formulae  fulfill  it  (except  Laspeyres'  and 
Paasche's,  which  are  also  odd  numbered).  This  makes  24  primary  for- 
mulae which  fulfill  the  proportionality  test. 

Table  57  gives  the  fulfillment  or  non-fulfillment  of  each  formula  as  to 
all  the  above  mentioned  tests  except  that  of  Commensurability  already  fully 
scored  in  the  paragraph  last  but  one.  In  the  table  a  "  X  "  signifies  ful- 
fillment and  a  "  —  "  signifies  non-fulfillment.  , 
From  this  table  it  is  clear  that  these  tests  differ  radically  from  the  re- 
versal tests  in  the  text  in  that  they  make  very  little  quantitative  discrimi- 

1  Thus  the  formulae  failing  to  fulfill  the  proportionality  test  are  not  true  averages,  except 
under  certain  conditions.  Such  a  formula  for  the  price  index  is  an  average  of  the  price 
relatives  only  when  the  quantity  relatives  are  all  equal. 


APPENDIX  I  421 

nation.  The  proportionality  test,  for  instance,  tells  us  that  certain  other 
formulae  do  agree  with  the  relatives  when  these  agree  with  each  other, 
which  agreement  is  certainly  to  their  credit.  Under  such  simple  circum- 
stances where  there  is  no  dispersion  all  these  various  index  numbers  agree 
with  each  other.  But  then,  no  index  number  is  needed !  When  there  is 
dispersion  this  test  disappears  and  the  various  index  numbers  scatter. 
That  is,  the  test  applies  when  we  do  not  need  its  help  and,  when  we  do,  it 
does  not  help  us. 

On  the  other  hand,  the  test  tells  us  that  certain  index  numbers  do  not 
exactly  agree  with  the  relatives  even  when  these  agree  with  each  other. 
This  is  certainly  to  their  discredit.  But,  from  a  practical  point  of  view, 
we  want  to  know  how  near  to  agreement  the  formula  then  comes.  We 
find  that,  in  some  cases,  the  disagreement  is  great  and,  in  others,  negligible 
so  that  the  mere  fact  of  non-agreement  is  of  little  practical  value. 

It  is  worth  while  to  note  that,  in  all  the  formulae  such  as  the  "super- 
lative" which  we  have  selected  on  other  grounds  as  superior  to  the  rest, 
the  proportionality  test  is  either  perfectly  fulfilled  or  almost  perfectly  ful- 
filled. That  is  to  say,  the  proportionality  test  never  conflicts  appreciably 
with  our  previous  conclusions  as  to  what  formulae  are  best,  although  it 
does  not  help  us  much  in  sifting  them  out  from  the  inferior  formulae.  It 
is  interesting  to  note  that  the  proportionality  test  shows  some  predilection 
for  the  aggregative  type  and  very  little  for  the  geometric.  This  is  in  spite 
of  the  fact  that  the  geometric  is,  par  excellence,  a  proportionality  type.  The 
reason  is  obviously  that  the  factor  antitheses  of  the  geometric  introduce 
a  discordant  element  —  the  value  ratio.  In  the  case  of  the  aggregative 
the  value  ratio  is  more  congenial.  Consequently,  of  the  index  numbers 
which  might  perhaps  be  called  the  two  chief  rivals  for  accuracy,  353  and 
5323,  the  former  conforms  to  the  proportionality  test  but  the  latter  does 
not  —  nor  does  5307,  the  best  of  the  arithmetic-harmonic  type.  Thus 
353  has  another  small  feather  in  its  cap.  In  fact  the  only  other  formulae, 
among  those  fulfilling  both  the  main  tests,  which  fulfill  also  the  proportion- 
ality test  are  1353,  2353,  and  3353,  all  aggregatives.  Thus  none  of  the 
others,  fulfilling  both  the  main  tests,  are,  strictly  speaking,  true  averages. 
As  to  the  determinateness  test  the  formulae  which  pass  this  test  perfectly 
are  usually  very  poor  formulae  while  many  of  the  best  ones  fail.  Formula 
353  and  all  the  aggregatives  pass;  but  307,  309,  323,  325,  5307,  5323  fail. 
Here  again  353  scores. 

As  to  the  withdrawal  and  entry  test,  it  follows  the  proportionality  test 
among  the  primary  formulae,  being  fulfilled  by  all  the  odd  numbered  for- 
mulae but  not  by  the  even  (except  those  which  are  also  odd).  But  when 
we  come  to  the  cross  formulae  few  meet  the  test. 

All  three  tests  relate  to  the  behavior  of  the  formula  under  some  special 
circumstances,  such  as  when  all  the  relatives  are  equal,  when  one  is  zero, 
or  when  one  coincides  with  the  index  number,  and  have  little  value  as  a 
general  guide.  All  the  good  formulae  which  fail  really  pass  practically. 

It  will  be  seen,  then,  that  those  three  tests  are  of  minor  importance.  This 
is  the  reason  I  have  not  made  use  of  them  in  the  text.  The  only  parts  of 
my  earlier  work  which  have  vital  importance  have  been  utilized  and  ampli- 
fied in  the  present  text.  These  three  minor  tests,  however,  while  weak, 


422 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  57.    SHOWING  THE  FORMULAE  WHICH  FULFILL  AND 
DO  NOT  FULFILL  THREE  SUPPLEMENTARY  TESTS 


FORMtTLA 

No. 

PROPOR- 
TIONALITY 

DETER- 

MI- 
NATENE8S 

WITH- 
DRAWAL 

AND 

ENTRY 

FORMULA 
No. 

PROPOR- 
TIONALITY 

DETER- 

MI- 

NATENES8 

WITH- 
DRAWAL 

AND 

ENTRY 

1 

X 

_ 

X 

201 

_ 

_ 

__ 

2 

— 

X 

— 

203 

X 

X 

— 

3 

X 

X 

X 

205 

X 

X 

— 

4 

X 

X 

X 

207 

— 

— 

— 

5 

X 

X 

X 

209 

— 

— 

— 

6 

X 

X 

X 

211 

— 

— 

— 

7 

X 

— 

X 

213 

— 

— 

— 

8 

_ 

X 

— 

215 

— 

— 

_ 

9 

X 

— 

X 

217 

X 

X 

— 

10 

— 

X 

— 

219 

X 

X 

— 

11 

X 

— 

X 

221 

— 

— 

— 

12 

— 

X 

— 

223 

— 

— 

— 

13 

X 

— 

X 

225 

— 

— 

— 

14 

— 

X 

— 

227 

— 

— 

_ 

15 

X 

— 

X 

229 

_ 

— 

_ 

16 

— 

X 

— 

231 

_ 

X 

_ 

17 

X 

X 

X 

233 

— 

X 

— 

18 

X 

X 

X 

235 

— 

X 

— 

19 

X 

X 

X 

237 

— 

X 

_ 

20 

X 

X 

X 

239 

— 

X 

— 

21 

X 

— 

X 

241 

— 

X 

— 

22 

— 

X 

— 

243 

— 

X 

_ 

23 

X 

— 

X 

245 

_ 

X 

_ 

24 

— 

X 

— 

247 

— 

X 

— 

25 

X 

— 

X 

249 

— 

X 

— 

26 

— 

X 

.— 

251 

— 

X 

— 

27 

X 

— 

X 

253 

X 

X 

— 

28 

— 

X 

— 

259 

X 

X 

— 

29 

X 

— 

X 

301 

— 

— 

— 

30 

— 

X 

— 

303 

X 

X 

— 

31 

X 

X 

X* 

305 

X 

X 

_ 

32 

— 

X 

— 

307 

— 

— 

— 

33 

X 

X 

X 

309 

— 

— 

— 

34 

— 

X 

— 

321 

— 

— 

— 

35 

X 

X 

X 

323 

_ 

_ 

— 

36 

— 

X 

_ 

325 

_ 

_ 

_ 

37 

X 

X 

X 

331 

_ 

X 

_ 

38 

— 

X 

— 

333 

— 

X 

— 

39 

X 

X 

X 

335 

— 

X 

_ 

40 

— 

X 

— 

341 

— 

X 

— 

41 

X 

X 

X 

343 

— 

X 

— 

42 

— 

X 

— 

345 

— 

X 

— 

43 

X 

X 

X 

351 

— 

X 

— 

44 

— 

X 

— 

353 

X 

X 

— 

45 

X 

X 

X 

1003 

X 

— 

X 

46 

•~ 

X 

—  ' 

1004 

— 

X 

~- 

*  In  case  of  withdrawal,  the  test  is  fulfilled  only  provided  (if  the  number  of  terms  is  first 
odd)  that  the  median  term  is  also  equal  to  the  median  of  the  two  neighboring  terms,  or  (if 
the  number  of  terms  is  first  even)  provided  the  two  middle  terms  are  equal.  Practically 
these  conditions  are  fulfilled,  at  least  approximately,  in  all  ordinary  circumstances.  In 
case  of  entry  no  such  reservations  are  necessary. 


APPENDIX  I 


423 


TABLE  57  (Continued) 


FORMTH.A 
No. 

PROPOR- 
TIONALITY 

DETER- 
MI- 
NATE NESS 

WITH- 
DRAWAL 

AND 

ENTRY 

FORMULA 
No. 

PROPOR- 
TIONALITY 

DETER- 

MI- 
NATENE8S 

WITH- 
DRAWAL. 

AND 

ENTRY 

47 

X 

X 

X 

1013 

X 

_ 

X 

48 

_ 

X 

_ 

1014 

— 

X 

— 

49 

X 

X 

X 

1103 

X 

— 

X 

60 

— 

X 

— 

1104 

— 

X 

— 

51 

X 

X 

X 

1123 

X 

— 

X 

52 

_ 

X 

_ 

1124 

_ 

X 

— 

53 

X 

X 

X 

1133 

X 

X 

X 

54 

X 

X 

X 

1134 

— 

X 

— 

59 

X 

X 

X 

1143 

X 

X 

X 

60 

X 

X 

X 

1144 

— 

X 

— 

101 

X 

— 

X 

1153 

X 

X 

X 

102 

— 

X 

— 

1154 

X 

X 

— 

103 

X 

X 

— 

1303 

— 

— 

— 

104 

X 

X 

— 

1323 

— 

— 

— 

105 

X 

X 

— 

1333 

— 

X 

— 

106 

X 

X 

— 

1343 

— 

X 

— 

107 

X 

— 

— 

1353 

X 

X 

— 

108 

— 

X 

— 

2153 

X 

X 

X 

109 

X 

— 

— 

2154 

X 

X 

—  • 

110 

— 

X 

— 

2353 

X 

X 

— 

121 

X 

— 

X 

3153 

X 

X 

X 

122 

— 

X 

— 

3154 

X 

X 

— 

123 

X 

— 

— 

3353 

X 

X 

— 

124 

— 

X 

_ 

4153 

X 

X 

X 

125 

X 

— 

— 

4154 

— 

X 

— 

126 

— 

X 

— 

4353 

— 

X 

— 

131 

X 

X 

X 

5307 

— 

— 

— 

132 

— 

X 

— 

5323 

— 

— 

— 

133 

X 

X 

— 

5333 

— 

X 

— 

134 

— 

X 

— 

5343 

— 

X 

_ 

135 

X 

X 

— 

6023 

X 

— 

X 

136 

— 

X 

— 

6053 

X 

X 

X 

141 

X 

X 

X 

8053 

X 

X 

— 

142 

— 

X 

— 

8054 

X 

X 

— 

143 

X 

X 

— 

8353 

X 

X 

— 

144 

— 

X 

— 

9001 

X 

— 

X 

145 

X 

X 

— 

9011 

X 

— 

X 

146 

— 

X 

— 

9021 

X 

— 

X 

151 

X 

X 

X 

9031 

X 

X 

X 

152 

— 

X 

— 

9041 

X 

X 

X 

153 

X 

X 

_ 

9051 

X 

X 

X 

154 

X 

X 

~ 

do  not  contradict  but  confirm  so  far  as  in  them  lies  the  conclusions  of  this 
book. 

Formula  353,  our  prize  formula  by  other  tests,  fulfills  perfectly  all  but 
one  of  these  three  minor  tests  and  fulfills  the  remaining  one  —  the  with- 
drawal and  entry  test  —  so  nearly  to  perfection  as  to  more  than  satisfy 
every  practical  demand. 

This  practical  fulfillment  would  be  clear  a  priori  even  if  we  were  to  make 
no  calculation  to  verify  it.  For  353  is  a  cross  between  53  and  54  each  oj 


424         THE  MAKING  OF  INDEX  NUMBERS 

which  fulfills  this  test  perfectly  and  which  are  always  close  to  each  other.  Fur- 
thermore, it  is  clear  that  a  newly  entered  commodity  the  price  relative 
for  which  in  1917  agrees  with  the  value  of  353  P  (as  it  was  prior  to  the 
entry  of  the  new  commodity)  both  being  161.558,  could  not,  if  its  weight 
or  importance  were  very  small,  disturb  appreciably  the  value  (161.558) 
which  353  already  had  while,  on  the  other  hand,  if  the  importance  of  the 
new  commodity  were  very  great,  i.e.  if  its  price  relative  were  heavily 
weighted,  it  would  so  dominate  the  index  number  as  to  make  its  value 
practically  coincide  with  its  own  (also  161.558).  Thus,  at  either  extreme, 
the  result  would  be  very  close  to  161.558 ;  and  it  stands  to  reason  that 
it  could  not  depart  from  this  very  much  at  intermediate  points. 

Formula  353  would  fulfill  this  test  at  all  intermediate  points  provided 

that  —  =  353Q.     That  is,  353P  would  remain  unchanged  by  entering  a 

new  commodity  such  that  2i  =  353P  provided  also  &  =  353Q.1 
PQ  qo 

If  the  ratio  to  be  entered,  — ,  is  not  equal  to  353Q  the  further  away  it  is 

qo 
the  more  will  the  new  353P  differ  from  the  old. 

To  take  an  example  more  extreme  than  any  met  with  even  among  our 
extremely  erratic  36  quantities,  let  q\  be  one  tenth  of  qQ.  Now  let  us  see 
how  far  353 P  can  get  from  fulfilling  the  withdrawal  and  entry  test,  by  (1) 
taking  the  case  (that  for  1917)  where  the  two  constituent  elements  53P 
and  54P  are  the  farthest  apart  and  (2)  assuming  that  while  the  price  ratio 

of  the  entered  commodity  agrees  with  353P  (  i.e.  —  =  1.61558 )  its  quantity 

V        flo  / 


ratio  is  absurdly  far  from  agreeing  with  353Q  (i.e.  —  = — ,  although 353Q, 

)\       q0     10 
. 

Let  po  =  1  and  pi  =  1.61558.  We  have  now  fixed  all  the  conditions 
except  the  absolute  values  of  qo  and  q\.  If  qo  is  very  small,  say  1  (and  so 
qi  is  .1),  the  effect  on  the  index  number  is  infinitesimal;  for,  before  the 
entry  of  p0  =  1,  pi  =  1.61558,  qQ  =  I  and  qi  =  .1  the  353P  was 


21238.49        25191.136 


13104.818        15641.85 
=  Vl.62066  X  1.61050  =  1.61558  j 
while  after  their  entry  353P  becomes 


Vspigo  + 
2 


1238.49  +  1.61558  X  1  x  25191.136  +  1.61558  X  .1  _ 
13104.818  +  1X1  15641.85  +  1  X  .1 

rCf.  Truman  L.  Kelley,  Quarterly  Publication  of  the  American  Statistical  Association, 
September,  1921,  p.  835.  The  apparently  different  formula  given  by  Professor  Kelley 
reduces  to  353Q. 


APPENDIX  I  425 


V1.62066  X  1.61050  =  1.61558 

the  ratio  of  which  to  the  original  1.61558  is  1.00000.     Evidently,  the  new 
figures  are  too  small  to  influence  the  result  appreciably. 

On  the  other  hand,  if  q0  is  very  large  (q\  being  always  ^  of  q0  and  p0 
being  1,  and  pi  being  1.61558),  say  g0  =  1,000,000,000,  the  result  is: 

1238.49  + 1.61558  X  1,000,000,000      25191.136  + 1.61558  X  100,000,000 
13104.818  +  1  X  1,000,000,000  15641.85  +  1  X  100,000,000 

=  Vl.61558  X  1.61558  =  1.61558 

the  ratio  of  which  to  1.61558  is  1.00000,  showing  that  the  new  figures  eclipse 
the  old  but  yield  the  same  result. 

Between  these  two  extremes  q0  has  a  value  which  makes  the  maximum 
discrepancy,  i.e.  which  renders  a  maximum,  or  minimum,  above  or  below 
unity,  the  ratio 


V 


P\q\ 


PoQo 


This  value  of  q0  is  obtained  by  differentiating  and  solving  for  g0  the 

equation  —  =  0. 
dqo 

Before  differentiating  we  may  omit  the  radical  sign  and  omit  the  denom- 
inator, for  the  ratio  R  is  a  maximum  or  minimum  according  as  its  square 
is  a  maximum  or  minimum,  which  in  turn  is  according  as  its  numerator  is 
a  maximum  or  a  minimum,  the  denominator  being  constant. 

For  simplicity  we  may  put  Spotfo  =  a,  2p0<?i  =  b,  *2p\qo  =  c,  Spi<?i  =  d. 
We  may  also,  for  convenience,  call  qQ  =  x  and  qi  =  kx  where  k  —  T&. 

Thus  we  are  to  maximize 


(c  +  pix\  /d  +  pi 
a  +  p&x)  \b  +  po 

or  (  substituting  m  =  —  ,  n  =  —  ,  r  =  —  ,  8  =  —  ) 

\  Pi         PQ          Pn          Pot/ 

to  maximize 


Differentiating  this  with  respect  to  x,  and  placing  the  result  equal  to 
zero,  we  have 

r)       (r  +  x}  (n  -  m} 


s  +  x  n  +  x 


426         THE  MAKING  OF  INDEX  NUMBERS 

Solving  for  x,  we  have 


where,  for  brevity 

g  -(*»+*)(«-  r)  +  («  +  r)(n  -  m) 
s  —  r  +  n  —  m 

r       mn  (s  —  r)  +  rs  (n  —  m) 
/i  =  -  • 
s  —  r  +  n  —  m 

It  remains  to  evaluate  x  numerically. 

The  result  of  solving  this  equation  is  x  =  qQ  =  45134.14  so  that  also 
poqo  —  45134.14  which  makes  the  new  index  number,  after  the  entry  of 
the  new  commodity,  1.61418  and  its  ratio  to  the  original  index  number, 
1.61558,  .99913,  instead  of  unity  as  it  is  if  qQ  is  very  small  or  very  large. 

In  other  words,  the  maximum  deviation  from  unity  occurs  when  the 
new  commodity  entered  has  a  value  in  1913  of  45134.14,  or  over  three  times 
the  total  value  (13104.818)  of  all  the  36  original  commodities.  Such  a 
gigantic  commodity  may  have  a  price  ratio  of  161.558  agreeing  with  the 
original  index  number  and  yet  its  entry  will  change  the  index  number  from 
161.558  to  161.418,  because  the  quantity  ratio  of  the  new  commodity  does 
not  agree  with  the  old  quantity  index,  being  .1  instead  of  1.1898.  Yet  even 
this  maximum  possible  wandering  from  161.558  is  negligible,  being  less 
than  one  part  in  a  thousand.  If  the  new  commodity  were  not  so  gigantic 
this  tiny  disturbance  would  be  much  tinier.  Thus  this  single  failure  of 
our  ideal  formula,  353,  to  fulfill  all  tests  applied  is  practically  not  a  failure. 

Note  to  Chapter  XIII,  §  10.  Ogburn's  Formula  for  Macaulay's  Theorem. 
Professor  W.  F.  Ogburn  has  derived  an  interesting  and  simple  formula  l 
for  the  difference  between  the  chain  and  fixed  base  index  numbers  when 
both  are  simple  arithmetics.  It  shows  that  we  may  always  tell  whether 
the  chain  or  fixed  base  figures  tend  to  be  the  greater,  by  watching  a  cri- 
terion. This  criterion  is  found  by  : 

(1)  Subtracting  any  price  relative  (say  that  of  bacon)  for  any  given  year 
from  the  index  number  of  that  year  ; 

(2)  Multiplying  the  difference  thus  found  by  the  percentage  increase  of 
the  price  of  that  commodity  (bacon)  between  said  year  and  the  next  ; 

(3)  Adding  the  product  thus  found  (which  may,  of  course,  be  positive  or 
negative)  for  bacon  to  the  corresponding  product  for  barley,  etc.,  through- 
out the  list. 

If  the  net  sum  thus  obtained  is  positive,  the  chain  figures  are  increasing 
(between  said  year  and  the  next)  faster  than  the  fixed  base  figures.  If  it 
is  negative,  the  opposite  is  true.  It  is  usually  positive  ;  because,  for  in- 
stance, the  lower  relatives,  affording  the  largest  differences,  are  the  most 
likely  to  recover  and  so  have  the  larger  percentage  increases  to  be  mul- 
tiplied by.  A  low  price  going  still  lower  is  the  exception. 

Note  to  Chapter  XIII,  §11.     If  a  Formula  Satisfies  the  Circular  Test  for 

*  See  Wesley  C.  Mitchell,  Buttetin  No.  884,  United  States  Bureau  of  Labor  Statistics, 
pp.  88-89,  footnote. 


APPENDIX  I  427 

Every  Three  Dates  It  Will  Satisfy  for  Four,  or  Any  Other  Number.  Thus, 
let  us  add  Boston  to  the  previous  trio  of  cities  (of  Chapter  XIII,  §  1)  and 
let  us,  in  thought,  step  the  price  levels  up  or  down  from  city  to  city  in  any 
desired  circuit  such  as  the  following :  Philadelphia,  New  York,  Boston, 
Chicago,  Philadelphia.  What  we  are  about  to  prove  is  that,  if  the  test  is 
fulfilled  for  every  triangular  comparison  among  these  four  cities,  it  will 
necessarily  be  fulfilled  for  the  quadrangular  comparison  stated. 

By  hypothesis  (i.e.  by  the  assumed  triangular  test)  we  know  that  passing 
around  the  triangle  Philadelphia,  Boston,  Chicago,  Philadelphia  we  re- 
turn to  the  same  figure,  100  per  cent,  with  which  we  started.  But,  by  the 
same  hypothesis  applied  to  a  different  triangle  of  cities,  we  know  that  the 
price  level  of  Boston,  calculated,  in  the  above  case,  directly  from  Phila- 
delphia, is  the  same  as  though  it  were  calculated  via  New  York.  Conse- 
quently, we  may,  without  affecting  the  result,  insert  New  York  between 
Philadelphia  and  Boston.  This  converts  the  original  triangular  circuit 
Philadelphia,  Boston,  Chicago  into  a  quadrangular  one,  Philadelphia, 
New  York,  Boston,  Chicago  without  disturbing  the  result,  namely,  that  we 
end  in  Philadelphia  at  the  same  figure  with  which  we  started. 

Algebraically,  we  wish  to  prove  that  PJ2  X  P2s  X  Pa*  X  P«  =  1,  and 
Pi2  X  P23  X  P34  X  P45  X  P5i  =  1,  etc.,  etc. 

Since   the   triangular  test  is  assumed  to  be  fulfilled,  we  know  that 

Pl2   X    P23   X   P31    =    1. 

But  for  P3i  we  may  substitute  P34  X  P4i,  —  since  the  triangular  test 

shows  that  P34  X  P«  X  PIS  =  1,  or  (since  P1S  =  -£-  )  P34  X  P«  =  PH. 

V  PSI/ 

Making  this  substitution,  we  have  Pi2  X  P23  X  P34  X  P«  =  1,  which 
is  the  proposition  to  have  been  proved,  —  for  four  steps  around  the  circle. 

Again,  substituting  in  the  last  for  P4i  the  expression  P46  X  P&\  we  have 
Pi2  X  P23  X  P34  X  P46  X  Psi  =  1  and,  substituting  likewise  for  P5i,  we 
have  Pi2  X  P23  X  P34  X  P45  X  Pw  X  P«  =  1,  etc.,  etc.,  which  were  to 
have  been  proved. 

Since  all  these  theorems  as  to  four,  five,  six,  etc..  years  follow  from  that 
for  three  only,  it  is  clear  that  the  essential  number  of  years  for  this  supposed 
test  is  three.  It  might,  therefore,  be  called  the  "triangular"  test  rather 
than  the  "circular"  test. 

In  other  words,  the  so-called  circular  test  really  starts  with  three  years. 
It  cannot  start  with  two  and  introduce  a  third,  fourth,  etc.,  on  the  analogy 
of  the  above  process,  as  the  reader  can  readily  convince  himself  if  he  tries 
it. 

Thus  the  triangular  test  is  on  a  different  plane  from  the  dual  or  time  re- 
versal test.  The  dual  test  befits  an  index  number  because,  by  its  very 
nature,  an  index  number  (such  as  Pi2)  involves  just  two  times,  such  as  "1 " 
and  "2,"  not  three.  The  triangular  test  introduces  an  extraneous  element 
not  already  represented  in  the  index  number  itself. 

Note  to  Chapter  XIV,  §  7.  Splicing  as  Applied  to  Aggregative  Index 
Numbers.  The  following  is  quoted  from  a  statement  kindly  sent  me  by 
Mr.  Charles  A.  Bell,  of  the  United  States  Bureau  of  Labor  Statistics,  show- 
ing the  method  of  splicing  employed  by  that  Bureau : 


428         THE  MAKING  OF  INDEX  NUMBERS 

In  general,  the  method  followed  by  the  Bureau  is  as  follows :  When  one  grade  or  quality 
of  an  article  is  to  be  substituted  for  another,  great  care  is  taken  that  the  newcomer  shall 
correspond  as  closely  as  possible  with  its  predecessor.  In  the  case  of  manufactured  prod- 
ucts, as  shoes  and  textiles,  the  manufacturer  furnishing  the  information  is  asked  to  make 
the  selection.  In  this  way  the  least  possible  violence  is  done  to  the  continuity  of  the  price 
series.  In  all  cases  of  this  kind  the  best  advice  available  is  sought.  The  two  series  are 
then  brought  together,  with  overlapping  data  for  at  least  one  full  year,  in  which  form  the 
detailed  price  information  is  published.  The  continuous  series  of  price  relatives  is  con- 
structed through  the  medium  of  the  overlapping  year,  which  carries  with  it  the  assumption 
that  prices  of  the  substituted  commodity  in  previous  years,  if  available,  would  have  shown 
the  same  degree  of  fluctuation  as  the  former  commodity. 

In  constructing  the  group  and  general  index  numbers,  the  plan  is  followed  of  building 
two  parallel  columns  of  weighted  price  aggregates  for  any  year  in  which  an  addition,  a  sub- 
stitution, or  a  withdrawal  takes  place.  The  first  column  contains  items  strictly  comparable 
with  those  for  preceding  years  and  the  second  column  contains  items  strictly  comparable 
with  those  for  succeeding  years.  The  index  number  for  the  overlapping  year  is,  of  course, 
based  on  the  items  in  the  first  column.  The  index  numbers  for  subsequent  years  are  found 
by  summing  the  items  for  such  years  and  converting  them  to  percentages  of  the  sum  in 
the  second  column  for  the  overlapping  year,  then  multiplying  them  by  the  index  number 
for  the  overlapping  year,  thus  converting  them  to  the  original  base.  This  is,  in  effect,  a 
chain  index  system,  welded  into  one  with  a  fixed  base.  Its  elasticity  permits  the  intro- 
duction or  dropping  of  commodities  without  serious  jar  to  the  structure,  although  the  effort 
is  made  to  reduce  to  a  minimum  consistent  with  fairness  the  number  of  changes  in  the  list 
of  commodities.  As  you  understand,  of  course,  the  Bureau  is  not  concerned  with  price 
relatives  of  individual  commodities  in  constructing  its  index  numbers. 

Note  to  Chapter  XIV,  §  9.  Bias  of  6023  and  23  Small  (in  the  case  of  the 
12  crops)  because  of  Correlation  between  Price  and  Quantity  Movements. 
There  is  another  reason  why  the  downward  bias  of  Formula  6023  is  so  small. 
This  is  that  the  downward  bias  of  23  itself  is  small.  This  is  because  of 
the  inverse  correlation  between  the  price  relatives  and  the  quantity  rela- 
tives. It  will  be  recalled  that  weight  bias  exists  in  a  price  index  because 
of  the  price  element  in  the  weight.  In  Formula  23,  for  instance,  the  index 
number  is  an  average  of  price  relatives  so  weighted  that  a  high  price 
relative  draws  a  low  price  element  in  its  weight  and  a  low,  a  high. 
The  other,  or  quantity  element,  was  assumed  as  likely  to  lean  in  one 
direction  as  the  other.  But  if  this  is  not  true ;  if,  instead,  every  high 
price  element  has  associated  with  it  a  low  quantity  element  and  vice 
versa,  evidently  the  weight  itself,  or  product  of  a  low  price  by  a  high  quan- 
tity, or  a  high  price  by  a  low  quantity,  will  be  devoid  of  bias.  If  the  price 
and  quantity  elements  are  thus  correlated  to  the  extreme  limit  of  100  per 
cent,  the  downward  bias  of  23  will  be  completely  abolished.  In  the  present 
case,  where  the  correlation  is  —  88  per  cent,  the  bias  is  nearly  abolished.  Were 
it  not  for  this  inverse  correlation  the  downward  bias  of  6023  (which  is  23 
with  broadened  base)  would  be  much  more  in  evidence. 

Note  to  Chapter  XV,  §  2.  Special  Proof  that  2153  is  Extremely  Close  to 
353.  Formula  2153  will,  under  all  ordinary  circumstances,  be  sufficiently 
close  to  Formula  353  to  serve  as  a  short  cut  substitute.  Only  where,  as  in 
this  monograph,  the  highest  accuracy  is  desired,  is  it  necessary  to  spend 
the  additional  time  for  calculating  Formula  353.  Formula  2153  may  be 
either  greater  or  less  than  353  according  to  circumstances.  It  is  desirable 
to  construct  a  table  by  which  we  may  know  how  close  2153  and  353  may  be 
under  various  circumstances.  The  two  formulae  (say,  for  prices,  which  we 
may  call  Formulae  2153P  and  353P)  will  coincide,  of  course,  if  Formulae 
53P  and  54P,  of  which  they  are  averages,  happen  to  coincide.  (In  this 


APPENDIX  I  429 

case,  Formulae  Nos.  53Q  and  54Q  will  also  coincide.)  The  two  (2153P 
and  353P)  will  also  coincide  if  53Q  and  54Q  happen  to  be  reciprocals  of 
each  other,  i.e.  if  one  of  the  latter  is  above  100  per  cent,  the  base,  and  the 
other  below  it  in  the  same  proportion.  In  all  other  cases,  2153P  and  353P 
will  differ. 

The  following  formula  1  gives  the  relative  size  of  2153P  and  353P : 


(2153P)  _  1  +  (54Q)   .      <54 
(353P)       1  +  (53Q)   '    \53' 


The  54  and  53,  under  the  radical,  may  be  either  both  "P's  "  or  both  "Q's," 
they  being  proportional.2 

The  reader  can  readily  verify  this  formula  by  substituting  in  it  the  ex- 
pression for  Formula  53,  etc.3 

From  this  formula,  it  follows  that  if 

<54P)/_(54gn 
(53P)  V      (53Q)/  " 
and  if,  furthermore, 

(54$)  X  (530)  >  1, 

then  2153P  will  exceed  353P  as  also  will  be  the  case  if  both  the  above  in- 
equalities are  reversed.     But  if  only  the  upper,  or  only  the  lower,  be  re- 
versed, then  2153P  will  be  less  than  353P. 
The  formula  may  also  be  written 


From  this,  knowing  f  f  and  353Q,  we  may  calculate  the  different  values 
of  the  formula  for  various  possible  values  of  f  f  and  353Q.  Evidently  if 
either  ff  or  353Q  is  equal  to  unity,  the  formula  reduces  to  unity.  That 
is,  if  either  (1)  53  and  54  are  close  together,  or  (2)  353Q  is  close  to  100  per 
cent,  then  2153  and  353  are  very  close  together. 

Table  58  tells  us  how  near  or  far  apart  are  Formulae  2153  and  353,  if  we 
know  (1)  how  near  or  far  apart  are  53  and  54,  and  (2)  how  large  or  small 
they  (and  their  average  353)  are. 


1  First  suggested   to   me,  in  substance,  by  Professor   Hudson  Hastings  of  the  Pollak 
Foundation  for  Economic  Research. 

2  By  definition  54P  =  V  4-  53Q,  likewise  54<?  -  V  -5-  53P ;  dividing  these  and  cancel- 
ing we  get  the  proportion.     (V  is  the  value  ratio.) 

8  He  may  also  be  interested  in  developing  the  formula?  (corresponding  somewhat  to  the 
above  for  2153),  for  2154,  2353,  8053, 8054,  8353,  in  terms  of  353  ;  also  in  terms  of  53  and  54. 
These  include  the  interrelations  connecting  all  available  types  of  averaging  the  two  for- 
mulae, 53  and  54,  i.e.  the  arithmetic  (8053),  harmonic  (8054),  geometric  (353),  and  aggre- 
gative (2153)  methods.  That  2153  is  an  aggregative  average  of  53  and  54,  i.e.  is 

numerator  of  Formula  53  +  numerator  of  Formula  54 

: — — — po         : — ,  is  clear  if  this  be  algebraically 

denominator  of  Formula  53  +  denominator  of  Formula  54 

expressed  and  compared  with  the  ordinary  formula  for  2153. 


430         THE  MAKING  OF  INDEX  NUMBERS 


TABLE  58.  FORMULA  2153P  AS  A  PERCENTAGE  OF  FOR- 
MULA 353P  (According  to  various  values  of  H  and  353Q,  both 
expressed  in  per  cents) 


FORMULA  353Q 

54 

53 

200 

150 

120 

100 

80 

50 

110 

101.6 

101.0 

100.4 

100.0 

99.5 

98.4 

105 

100.8 

100.5 

100.2 

100.0 

99.7 

99.2 

102 

100.3 

100.2 

100.1 

100.0 

99.9 

99.7 

100 

100.0 

100.0 

100.0 

100.0 

100.0 

100.0 

98 

99.7 

99.8 

99.9 

100.0 

100.1 

100.3 

95 

99.1 

99.5 

99.8 

100.0 

100.3 

100.9 

90 

98.3 

99.0 

99.5 

100.0 

100.6 

101.8 

From  this  table  it  will  be  seen  that  the  index  number  by  Formula  2153 
is  always  close  to  that  by  353,  even  under  the  extreme  conditions  repre- 
sented by  the  four  corners  of  the  table  —  conditions  seldom,  if  ever,  realized 
in  practice.  The  upper  left  corner  represents  a  condition  where  £f  is 
110  per  cent,  i.e.  where  Formula  54  exceeds  53  by  10  per  cent  (a  difference 
probably  never  reached  in  practice)  combined  with  the  additional  fact 
that  the  price  level  is  very  high  (200  per  cent).  Under  these  two  circum- 
stances, the  ratio  of  2153P  to  353P  is  101.6,  i.e.  2153P  is  1.6  per  cent  higher 
than  353P. 

In  the  other  three  corners  other  extreme  circumstances  are  represented. 
The  table  shows  that,  even  if  only  one  of  the  two  conditions  is  extreme,  the 
two  index  numbers,  2153  and  353,  coincide  as  perfectly  as  when  neither 
is  extreme.  By  means  of  this  table,  it  is  easy  to  tell  in  any  individual  case 
how  great  an  error  will  be  involved  by  using  2153  instead  of  353,  and 
whether  the  additional  accuracy  of  353  is  worth  the  additional  trouble. 
In  the  case  of  the  36  commodities,  there  is  no  instance  where  353  would 
be  needed  as  2153  is  close  to  353,  always  within  one  tenth  of  one  per  cent. 
The  reason  is  that  53  and  54  are  so  close  together.  In  the  case  of  Persons' 
statistics  for  12  crops,  Formulae  53  and  54  are  further  apart.  But  even 
this  fact  does  not  require  the  use  of  Formula  353  except  possibly  in  the  case 
of  the  year  1890,  where,  besides  the  fact  that  if  is  low  (94.85  per  cent), 
there  is  the  additional  fact  that  353Q  is  very  low  (56.7  per  cent).  In 
this  case,  the  ratio  of  Formula  2153P  to  353P  is  100.8  per  cent.  That  is, 
the  two  differ  by  three  fourths  of  one  per  cent.  This  is  the  greatest  error 
I  can  find  in  any  actual  case  and  this,  in  most  cases,  would  not  be  con- 
sidered worth  taking  into  account. 

Note  to  Chapter  XVI,  §4.  "Probable  Error"  by  Professor  Kelley's 
Method.  Professor  Truman  L.  Kelley  *  proposes  another  method  of 

1  "Certain  Properties  of  Index  Numbers,"  Quarterly  Publication  of  the  American  Statis- 
tical Association,  pp.  826-41,  September,  1921, 


APPENDIX  I  431 

measuring  the  "probable  error"  of  an  index  number,  meaning  the  error 
due  to  incompleteness  of  sampling,  or  smallness  of  the  number  of  commodi- 
ties included  in  the  index.  His  method  is  to  divide  the  list  of  commodities 
into  halves,  calculate  (by  the  same  formula  as  that  used  for  the  entire  set)  the 
series  of  index  numbers  for  each  of  the  halves,  take  the  coefficient  of  corre- 
lation, r,  between  these  two  series  of  index  numbers,  or  "sub-indices,"  take 

their  "reliability  coefficient,"  R,  which  is  equal  to  — — ,  take  the  standard 

deviation  of  each  of  the  two  series  of  sub-indices  (from  the  mean  of  the 
series),  take  the  average  <r  of  these  two  and,  from  this,  calculate  the  stand- 
ard deviation  of  the  original  index,  for  the  same  period,  by  the  formula 

a'  =<TA/          .     Having  thus  obtained  R  and  o-',  he  obtains  the  desired 
\     2 

"probable  error"  of  the  original  set,  by  the  formula  l 

P.  E.  =  .6745<r'  Vl  -  R. 

Applying  this  formula  to  our  200  commodities,  we  find,  after  dividing 
them  into  two  groups,  A  and  B,  of  100  each,  selected  by  lot,  that  the  stand- 
ard deviation  of  A  is  .0344,  and  of  B.  .0351  giving  a  =  -0344  +  -0351  = 

.03475  and  r  =  .790 ;  whence  a'  -  .0329  and  R  =  .883  and  P.  E.  =  .008. 

That  is,  according  to  this  reckoning,  the  200  commodities  considered 
as  samples  give  an  index  number  the  probable  error  of  which,  in  the  sense 
of  its  deviation  from  an  ideally  complete  set  of  commodities,  is  .008,  or  a 
little  less  than  1  per  cent. 

But  the  two  100  lists,  A  and  B,  differ  from  the  200  list  in  not  being  in- 
tentionally selected  as  good  samples.  In  the  following  example,  the  200 
list  is  divided  into  two  100  lists  by  a  mixture  of  lot  and  assorting  such 
that,  so  far  as  possible,  A'  and  B'  are  equally  well  assorted  as  samples  and 
have  equal  importance  or  weights.  We  find  the  standard  deviation  of  A '  is 

.0432,  of  B'  .0316  giving  a  =  -0432  +  -0316  =  .0374  and  r  =  .501 ;  whence 

a'  =  .0324  and  R  =  .668  and  P.  E.  =  .013,  or  1.3  per  cent,  a  result  very 
close  to  the  H  per  cent  by  my  own  method. 

The  fact  that  the  former  and  more  completely  random  application  of 
Kelley's  method  gives  a  smaller  result  may,  I  think,  properly  be  called 
accidental.  We  would  expect  the  opposite  contrast. 

Professor  Kelley  warns  against  using  his  method  when  the  dates  for  the 
quotations  are  too  close  together.  "It  is  desirable  that  the  time  interval 
between  successive  indices  be  sufficient  to  insure  the  relative  independence 
of  the  commodity  quotations  involved."  2  This  seems  to  me  to  constitute 
a  serious  weakness  in  the  method,  a  weakness  which  does  not  apply  to  the 
method  in  the  text.  In  the  present  case  the  time  intervals  are  short,  aver- 
aging less  than  three  months. 

1  He  also  gives  (p.  832)  a  special  formula  for  the  probable  error  in  the  case  of  geometric 
formulae  (our  Formulae  21  and  9021). 
*  Ibid.,  p.  830. 


432         THE  MAKING  OF  INDEX  NUMBERS 

Note  to  Chapter  XVI,  §  7.  Round  Weights  for  the  Majority  of  Commodi- 
ties are  Sufficiently  Accurate.  The  proof  is  as  follows :  First  compute  the 
index  number  as  proposed,  i.e.  with  statistical  weights  for  the  most  impor- 
tant 28,  and  the  round  weights  nearest  thereto  for  the  172  others.  Thus, 
for  wheat  No.  2,  red,  the  statistical  quantity  603  was  used  and  multiplied 
by  the  price  at  any  time.  But  for  citric  acid  the  quantity  is  given  statisti- 
cally as  3.36  but  the  quantity  used  in  my  index  number  is  the  nearest  round 
number,  10,  this  being  nearer  than  I.1  Similarly  for  turpentine,  the  quan- 
tity statistically  given  is  53  gallons  instead  of  which  we  use  100,  the  nearest 
round  number.  After  doing  likewise  for  each  of  the  172  commodities  we 
calculate  the  index  number  for  the  200  commodities. 

Having  obtained  this  index  number  by  using  the  nearest  round  weights, 
we  next  compare  it  with  what  it  would  be  if  the  exact  weights  had  been 
used.  The  two  differ  by  less  than  one  per  cent  even  when  the  dispersion 
of  prices  is  as  great  as  for  1916  relative  to  1913,  a  dispersion  seldom  reached 
inside  of  40  years  (as  shown  in  Table  10  in  Chapter  V).  We  may  therefore 
rely  on  this  short  cut  method  to  give  results  within  one  per  cent  of  what  the 
long  method  would  give.  This  error  of  less  than  one  per  cent  is  the  error 
of  any  index  number  relative  to  the  base.  The  error  from  month  to  month 
would,  of  course,  be  still  less. 

Note  to  Chapter  XVII,  §  14.    List  of  Calculated  Index  Numbers 
1.   Discontinued  Index  Numbers 

Ferguson,  Roman  Empire  (301  A.D.)  ;  Leber,  France  (900-1847); 
Shuckburgh  Evelyn,  Great  Britain  (1050-1800);  d'Avenel,  France 
(1200-1790);  Rogers,  Great  Britain  (1259-1793);  Hanauer,  France 
(1351-1875);  Vaughan,  Great  Britain  (1352-1650);  Wiebe,  Great 
Britain  (1451-1600);  Dutot,  France  (1462-1715);  Wiebe,  France 
(1493-1600) ;  Gilliodts,  Belgium  (1500-1600) ;  Carli,  Italy  (1500-1750) ; 
Elmes,  Great  Britain  (1600-1800);  Jevons,  Great  Britain  (1782-1865); 
Roelse,  United  States  (1791-1801);  Flux,  Great  Britain  (1798-1869); 
Hansen,  United  States  (1801-1840);  Hurlin,  United  States 
(1810-1920);  Burchard,  United  States  (1825-1884);  Juergens, 
United  States  (1825-1863) ;  de  Foville,  France  (1827-1880) ;  Laspeyres, 
Germany  (1831-1863);  Porter,  Great  Britain  (1833-1837);  Walker, 
United  States  (1834-1859) ;  Giffen,  Great  Britain  (1840-1883) ;  Falkner 
(Aldrich  Senate  Report),  United  States  (1840-1891);  Mulhall,  Great 
Britain  (1841-1884);  Krai,  Germany  (1845-1884);  Bourne,  Great 
Britain  (1845-1879) ;  Levasseur,  France  (1847-1856) ;  Paasche,  Germany 
(1847-1872) ;  Soetbeer,  Germany  (1847-1891) ;  Denis,  Belgium  (1850- 
1910) ;  Schmitz,  Germany  (1851-1913)  ;  Drobisch,  Germany  (1854-1867) ; 
Ellis,  Great  Britain  (1859-1876) ;  Mitchell,  Germany,  Great  Britain,  and 
United  States  (1860-1880) ;  Wasserab,  Germany  (1861-1885) ;  Atkinson, 
India  (1861-1908);  Mcllraith,  New  Zealand  (1861-1910);  Powers, 
United  States  (1862-1895) ;  Palgrave,  Great  Britain  and  France  (1865- 
1886) ;  Jankovich,  Austria  (1867-1909) ;  Daggett,  United  States  (1870- 

1  The  half-way  point  between  1  and  10  is  best  taken  as  \/l  X  10  =  3.16,  rather  than  as 
j  (1  +  10)  or  5.5,  although  the  difference  between  the  results  of  using  3.16  or  5.5  is  negligible. 


APPENDIX  I  433 

1894);  Walras,  Switzerland  (1871-1884);  van  der  Borght,  Germany 
(1872-1880) ;  Fisher  (from  Japanese  Report  of  the  Commission  for  In- 
vestigation of  Monetary  Systems),  China,  India,  and  Japan  (1873-1893, 
except  China  which  commenced  in  1874) ;  Flux,  France  (1873-1897) ; 
Hansard,  Great  Britain  (1874-1883) ;  von  Inama-Sternegg,  Austria 
(1875-1888) ;  Koefoed,  Denmark  (1876-1919) ;  Bureau  of  Economic  Re- 
search, United  States  (1878-1900) ;  Kemmerer,  United  States  (1879-1908) ; 
Julin,  Belgium  (1880-1908);  Levasseur,  France  (1880-1908);  Conrad, 
Germany  (1880-1897) ;  Einar  Rudd,  Norway  (1880-1910) ;  Waxweiler, 
Belgium  (1881-1910) ;  Nicolai,  Belgium  (1881-1909) ;  Sauveur,  Belgium 
(1881-1909) ;  Zahn,  Germany  (1881-1910) ;  Zimmerman,  Germany  (1881- 
1910);  Falkenburg,  Netherlands  (1881-1911);  Methorst,  Netherlands 
(1881-1911);  Alberti,  Italy  (1885-1911);  Hartwig,  Germany  (1886-1910) ; 
O'Conor,  India  (1887-1902) ;  Eulenberg,  Germany  (1889-1911) ;  Hooker, 
Germany  (1890-1911);  Datta  and  Shirras,  India  (1890-1912);  Imperial 
Ministry  of  Commerce  and  Industry,  Petrograd,  Russia  (1890-1912) ; 
La  Re*forme  ficonomique,  France  (1891-1913);  Flux,  Germany  (1891- 
1897) ;  Bernis,  Spain  (1891-1913) ;  Barker,  United  States  (1891-1896) ; 
Calwer,  Germany  (1895-1909);  Fisher,  United  States  (1896-1918); 
Vossische  Zeitung,  Germany  (1900-1912) ;  Loria,  Italy  (1900-1909) ; 
Ottolenghi,  Italy  (1910-1918) ;  Pearl  (U.  S.  Food  Administration),  United 
States  (1911-1918) ;  Statistical  Department  of  Stuttgart,  Germany  (Stutt- 
gart) (1913-1919) ;  Mitchell  (War  Industries  Board),  United  States  (1913- 
1918);  Foster,  United  States  (1913-1919);  Statistical  Department  of 
Nurnberg,  Germany  (Nurnberg)  (1914-1920). 

2.  Current  Index  Numbers 

Argentina:  Revista  de  Economia  Argentina,  Bunge,  wholesale  (im- 

ports and  exports). 

Ibid.,  Bunge,  retail,  18  commodities,  Formula  9001. 
Ibid.,  Bunge,  cost  of  living,  Formula  9001. 

Australia:  Quarterly   Summary   of   Australian   Statistics,    Knibbs, 

(Melbourne)     wholesale,  92  commodities,  Formula  53. 

Quarterly  Statistical  Bulletin  of  New  South  Wales,  whole- 
sale, 100  commodities. 

Quarterly   Summary    of   Australian   Statistics,    Knibbs, 
cost  of  living,  46  commodities  and  rent,  Formula  53. 

Austria :  Mitteilungen  des  Bundesamtes  fur  Statistik,  Bundesamt 

(Vienna)  fur  Statistik,  retail,  23  commodities. 

Ibid.,  Paritdtische  Kommission,  cost  of  living,  23  com- 
modities. 

Belgium :  Department  of  Statistics,  wholesale,  130  commodities. 

Revue  du  Travail,  wholesale,  209  commodities  (more  or 

less  from  time  to  time),  Formula  21. 

Ibid.,  wholesale,  127  commodities,  Formula  21. 

Ibid.,  retail,  22  commodities,  Formula  9001. 

Ibid.,  retail,  30  commodities,  Formula  9001. 

Ibid.,  cost  of  living,  56  commodities,  Formula  1. 


434         THE  MAKING  OF  INDEX  NUMBERS 


Bulgaria :  Bulletin  statistique  mensuel  de  la  Direction  Generate  de  la 

Statistique,  wholesale. 
Ibid.,  retail,  47  commodities,  Formula  3. 

Canada :  Labour  Gazette,  Coats,  wholesale,  238  commodities,  For- 

mula 53. 

Ibid.,  Coats,  cost  of  living,  29  staple  foods,  5  fuel  and 
light,  clothing,  rent,  and  sundries. 

Federal   Reserve  Bulletin,  wholesale,    101   commodities, 
Formula  53. 

Monthly  Commercial  Letter,  Canadian  Bank  of  Commerce, 
Michell,  wholesale,  48  commodities. 
Toronto  newspapers,  Michell,  wholesale,  40  commodities, 
Formula  1. 
China:  Finance  and  Commerce  (Shanghai),  Bureau  of  Markets, 

(Shanghai)        Treasury  Department,  wholesale,  147  commodities. 
Czechoslovakia:  Monthly  Price   Bulletin,  Statistical  Office,  Ryba,   retail, 

25  commodities,  Formula  1. 
Denmark:  Finanstidende,  wholesale,  33  commodities,  Formula  9001. 

Statistiske  Efterretninger,  cost  of  living. 

Dutch  East          Statistical  Bureau  of  the  Department  of  Agriculture,  whole- 
Indies  :  sale. 

Egypt:  Monthly  Agricultural   Statistics,   Statistical    Department, 

(Cairo)  wholesale,  26  commodities,  Formula  21. 

Ibid.,  retail,  23  commodities,  Formula  9001. 
Ibid.,  cost  of  living. 
Finland :  Social  Tidskrift,  cost  of  living,  17  commodities,  rent,  fuel, 

a  daily  newspaper,  and  taxes. 

France :  Bulletin  de  la  Statistique  Ginerale  de  France,  March,  whole- 

sale, 45  commodities,  Formula  1. 
Ibid.,  March,  retail,  13  commodities,  Formula  53. 
(Paris)  Ibid.,  March,  cost  of  living,  13  commodities,  Formula  53. 

Germany :  Frankfurter  Zeitung,  wholesale,  98  commodities,  Formula  1 . 

Wirtschaft  und  Statistik,  Statistisches  Reichsamt,  whole- 
sale, 38  commodities,  Formula  3. 

(Halle)  Statistische  Vierteljahrshefte,  Statistisches  Ami    der  Stadt 

Halle,  retail,  41  commodities. 

Wirtschaft  und  Statistik,  Statistisches  Reichsamt,  cost  of 
living,  17  commodities  and  rent. 

•Monatliche    tJbersichten  tiber  Lebensmittelpreise,   Calwer, 
cost  of  living,  19  commodities. 

(Berlin)  Finanzpolitische  Korrespondenz,  Kuczynski,  minimum  cost 

of  living,  19  commodities,  rent,  and  miscellaneous,  For- 
mula 53. 
(Berlin)  Die  Kosten  des  Erndhrungsbedarfs,  Silbergleit,  cost  of  living 

(food),  Formula  1. 

(Frankfurt-       Indexziffem  (published  by  Reitz  and  Kohler,  Frankfurt- 
am-Main)     am-Main) ,  Elsas,  cost  of  living,  40  commodities,  Formula  3. 
(Hannover)      Mitteilungen  des  Statistischen  Amts  der  Stadt  Hannover, 
cost  of  living,  37  commodities,  Formula  9001. 


APPENDIX  I 


435 


(Kb'ln)  Statistische   Monatsberichte,    Statistisches    And,    cost    of 

living. 

(Leipzig)  Statistisches  Amt,  cost  of  living. 

(Ludwigs-         Statistische   Vierteljahrsberichte  der  Stadt  Ludwigshafen, 

hafen)  cost  of  living. 

(Mannheim)     Mannheimer  Tageszeitung,  Hofmann,  cost  of  living,  79 

commodities,  rent,  and  miscellaneous. 

Great  Britain :      Board  of  Trade  Journal,  Flux,  wholesale,  150  commodi- 
ties, Formula  21. 

Economist,  wholesale,  44  commodities,  Formula  1. 

Federal  Reserve  Bank  of  New    York,  Monthly  Review, 

Snyder,  wholesale,  20  commodities. 

Federal    Reserve    Bulletin,   wholesale,   98   commodities, 

Formula  53. 

Statist,  wholesale,  45  commodities,  Formula  1. 

Times   (London),   Crump,   wholesale,   70  commodities, 

Formula  21. 

Labour  Gazette,  cost  of  living,  41  commodities  and  rent, 

Formula  9001. 

Hungary :  Szakszervezeti  Ertesito,  cost  of  living,  34  commodities. 

India :  Labour  Gazette,  Shirras,  wholesale,  43  commodities,  For- 

(Bombay)         mula  1. 

(Calcutta)          Department  of  Statistics,  wholesale,  75  commodities. 
(Bombay)          Labour  Gazette,  Shirras,    cost  of  living,  23  commodities 

and  rent. 
Italy :  Annuario  Statistico  Italiano,  wholesale,  13  commodities, 

Formula  1. 

L' Economista,     Bachi,     wholesale,     100     commodities, 

Formula  1,  chain,  Formula  21. 

La  Riforma  Sociale,  Necco,  wholesale  (imports  and  ex- 
ports), 19  imports  and  12  exports. 

(Milan)  Bollettino  municipale  mensile,  cost  of  living. 

(Rome)  Bollettino  del  Ufficio  del  Lavoro,  cost  of  living. 

(Florence)          Ufficio  di  Statistica,  cost  of  living. 

Japan :  Bank  of  Japan,  wholesale,  56  commodities,  Formula  1. 

(Tokio)  Department  of  Agriculture    and    Commerce,   wholesale, 

39  commodities. 

Oriental  Economist,  wholesale. 
Netherlands:       Maandschrift  van  het  Centraal  Bureau  voor  de  Statistiek, 

wholesale,  53  commodities,  Formula  1. 
(Amsterdam)    Maandbericht  van  het  Bureau  van  Statistiek,  retail,  26 

commodities,  Formulae  1  and  3. 

Ibid.,  cost  of  living. 

(Hague)  Maandcijfers  van  het  Statistish  Bureau,  cost  of  living. 

New  Zealand :      Monthly  Abstract  of  Statistics,   Fraser,   wholesale,    140 

commodities,  Formula  53. 

Ibid.,  Fraser,  cost  of  living,  66  commodities  and  rent. 

Ibid.,  Fraser,  export  prices. 

Ibid.,  Fraser,  producers'  prices. 


436 


THE  MAKING  OF  INDEX  NUMBERS 


Norway :  Oekonomisk  Revue,  wholesale,  70  commodities,  Formula  1. 

Farmand,  wholesale,  40  commodities,  Formula  1. 

Statistiske  Meddelelser,  Dei  Statistiske  Centralbyra,  cost  of 

living,  Formula  53. 
Peru  Direction   de    Estadistica,    wholesale,    58    commodities, 

Formula  1. 

Poland :  Central  Statistical  Office,  wholesale,  68  commodities,  For- 

mula 21. 
(Warsaw)         Statystyka  Pracy  of  the  Central  Statistical  Office,  cost  of 

living,  38  commodities  and  rent. 
Russia :  Ekonomicheskaia  Zhizn,  retail,  22  commodities. 

(Moscow) 
South  Africa :       Quarterly  Abstract  of    Union  Statistics,  wholesale,    188 

commodities,  Formula  53. 

Ibid.,  Cousins,  retail,  23  commodities,  Formula  53. 

Ibid.,  cost  of  living,  19  commodities  and  rent,  Formula 

53. 

Spain :  Instituto  Geografico  y  Estadistico,  wholesale,  74  commodi- 

ties, Formula  1. 

Ibid.,  retail,  28  commodities. 

(Barcelona)       Bulleti  del  Museo  Social,  wholesale,  25  commodities. 
Sweden :  Goteborgs  Handels-och  Sjofartstidning,  Silverstolpe,  whole- 

sale, 47  commodities,  Formula  53. 

Kommersiella  Meddelanden,  wholesale,  160  commodities, 

Formula  3. 

Sociala  Meddelanden,  cost  of  living,  75  commodities,  rent, 

taxes,  and  miscellaneous. 
Switzerland :         Neue  Zurcher  Zeitung,  Lorenz,  wholesale,  71  commodities, 

Formula  9001. 
(Basle)  Statistische  Monatsberichte,  retail,  21  commodities. 

Schweizerischer  Konsumverein,  retail,  41  commodities. 
-(Berne)  Halbjahrsberichte  des  Statistischen  Amis  der  Stadt  Bern, 

retail,  79  commodities. 

(Zurich)  Statistik  der  Stadt  Zurich,  cost  of  living. 

United  States :     Annalist,  wholesale,  25  commodities,  Formula  1. 

Bradstreet,  wholesale,  96  commodities,  Formula  51. 

Babson,  wholesale,  10  commodities,  Formula  1. 

Bureau  of  Labor  Statistics,  Monthly  Labor  Review,  Stewart, 

wholesale,  404  commodities  (more  or  less  from  time  to 

time),  Formula  53. 

Dun's  Review,  Little,  wholesale,  about  300  commodities, 

Formula  53. 

Federal   Reserve  Bulletin,   wholesale,    104   commodities, 

Formula  53. 

Federal  Reserve  Bank  of  New    York,  Monthly  Review, 

Snyder,  wholesale,  20  commodities. 

Gibson's  Weekly  Market  Letter,  wholesale,  22  commodities. 

Harvard   Review  of  Economic  Statistics,  Persons,  whole- 
sale, 10  commodities,  Formula  21. 


APPENDIX  I  437 

San  Diego  (California)  Union,  Bissell,  wholesale,  60 
commodities,  Formula  21. 

Bulletin,  National  City  Bank  of  New  York,  Austin, 
wholesale  (imports  and  exports),  25  imports  and  30  ex- 
ports, Formula  51. 

Bureau  of  Labor  Statistics,  Monthly  Labor  Review,  Stewart, 
retail,  43  commodities,  Formula  53. 
Ibid.,  Stewart,  cost  of  living,  184  commodities  and  rent. 
Massachusetts  Special  Commission  on  the  Necessaries  of 
Life,  Parkins,  cost  of  living,  78  commodities,   Formula 
9001. 

National  Bureau  of  Economic  Research,  King,  cost  of  liv- 
ing for  families  spending  $25,000  per  annum,  Formula 
9001. 

National  Industrial  Conference  Board  Monthly  Service 
Letters  and  Reports,  Stecker,  cost  of  living,  90  items  and 
rent,  Formula  53. 

Federal  Reserve  Bulletin,  agricultural  movements,  14  com- 
modities, Formula  53. 

Ibid.,  mineral  production,  7  commodities,  Formula  53. 
Ibid.,  manufactured  goods,  34  commodities,  Formula  53. 
Harvard  Review  of  Economic  Statistics,  volume  of  pro- 
duction (agriculture),  12  commodities,  Formula  6023. 
Ibid.,  volume  of  production  (mining),  Day,  9  commodi- 
ties, Formula  6023. 

Ibid.,  volume  of  production  (manufacture),  Day,  33  series. 
Ibid.,  volume  of  production  (last  3  combined),  Day. 
Ibid.,   Aberthaw,   cost   of   reinforced   concrete  factory 
building. 

Summary  of  Business  Conditions  in  the    United  States, 
Am.  Tel.  &  Tel.  Co.,  construction  costs,  15  principal 
building  materials  and  weighted  average  of  wage  rates. 
Fred  T.  Ley  &  Co.  (Springfield,  Mass.},  cost  of  building 
construction. 

American   Writing   Paper  Company,  paper  production 
costs,  5  materials  and  labor,  Formula  1. 
Federal   Reserve   Bulletin,    foreign   exchange    rates,     18 
leading  currencies,  Formula  29.     (For  other  such  indexes 
—  English,  German,  Swedish,  Norwegian  —  see  Federal 
Reserve  Bulletin,  July,  1921,  p.  794.) 
Annalist,  stocks,  25  railroads  and  25  industrials. 
New  York  Times,  stocks,  50. 
Wall  Street  Journal,  stocks,  20  railroads. 
Many  other  trade  journals  and  newspapers  carry  index 
numbers  of  stocks  or  bonds,  or  both. 

For  fuller  information  on  many  of  the  above  index  numbers,  see  Bulletin 
284,  United  States  Bureau  of  Labor  Statistics ;  International  Labour  Re- 
view, pp.  52-75,  July,  1922 ;  and  Emil  Hofmann,  Indexziffern  im  Inland 


438          THE  MAKING  OF  INDEX  NUMBERS 

und  im  Ausland,  127  pp.  G.  Braunsche  Hofbuchdruckerei  und  Verlag, 
Karlsruhe,  1921. 

The  above  list  is  exclusive  of  index  numbers  of  wages  and  of  a  great  many 
index  numbers  bearing  on  prices,  the  cost  of  living,  etc.  as  between  different 
places.  For  information  as  to  index  numbers  of  wages  the  reader  is  referred 
to  the  United  States  Bureau  of  Labor  Statistics,  the  International  Labour 
Office,  and  the  National  Industrial  Conference  Board.  For  information 
with  regard  to  place  to  place  index  numbers,  see  also  Report  of  an  Enquiry 
by  the  Board  of  Trade  (British)  into  Working  Class  Rents,  Housing  and  Retail 
Prices,  1911. 

In  addition  to  the  above  specific  index  numbers  various  attempts  have 
been  made  to  use  index  numbers  of  index  numbers,  or  averages  of  aver- 
ages. For  example,  George  H.  Wood l  undertook  to  express  the  develop- 
ment of  the  consumption  of  the  English  population,  and  Neumann- 
Spallart  to  find  a  "  measure  of  the  variations  in  the  economic  and  social 
condition  of  nations"  by  "mean  index  numbers."2  We  might  also 
include  under  the  rubric  of  index  numbers  the  various  trade  barometers, 
etc.,  which  are  in  commercial  use,  such  as  Brookmire's,  Babson's,  the 
Harvard  Committee  on  Economic  Research,  the  Alexander  Hamilton 
Institute,  the  American  Institute  of  Finance,  the  Standard  Statistics 
Corporation,  the  London  School  of  Economics,  etc. 

1  George  H.  Wood,  "Some  Statistics  of  Working  Class  Progress  since  1860."    Journal 
of  the  Royal  Statistical  Society,  p.  639  et  seq.,  esp.  p.  654  et  seq. 

2  See  Franz  %zek,  Statistical  Averages  (translated  by  Warren  M.  Persons),  New  York, 
1913,  pp.  95-101,  esp.  p.  100. 


APPENDIX  II 

THE  INFLUENCE  OF  WEIGHTING 

§  1.  Introduction 

The  "best  method  of  weighting"  index  numbers  has  long  been  the  sub- 
ject of  debate.  We  have  seen,  however,  that  any  method  which  is  really 
systematic,  —  whether  it  be  7,  //,  ///,  IV,  or  one  of  the  cross  weight  sys- 
tems, —  can  be  used  to  start  with,  provided  the  index  number  so  obtained  is 
subsequently  rectified.  Rectification  will  take  out  the  bias,  however 
great  it  may  be  to  start  with.  Only  freakish  weighting  is  incorrigible. 

Consequently,  the  whole  subject  of  "the  proper  weighting"  really  dis- 
appears in  the  result  and  plays  no  part  in  the  main  argument  of  this  book. 
But  in  view  of  the  literature  on  the  subject  and  in  order  to  effect  an  ad- 
justment between  current  ideas  and  the  conclusions  of  this  book,  the  sub- 
ject is  included,  though  relegated  to  this  Appendix  so  as  not  to  interrupt 
the  main  course  of  reasoning  in  the  text.  In  a  few  instances  we  shall  need 
to  repeat  slightly  some  of  the  observations  in  the  text. 

We  began  with  a  discussion  of  "simple"  index  numbers.  These  are 
often  loosely  referred  to  as  "unweighted"  index  numbers.  More  properly, 
of  course,  they  are  evenly  weighted  index  numbers,  i.e.  index  numbers  in 
which  every  price  relative  has  the  same  weight  as  every  other. 

We  next  noted  (for  all  types  of  index  numbers  except  the  aggregative) 
four  methods  of  weighting  by  values,  viz.  I  (by  values  of  the  commodities 
in  the  base  year) ;  7  V  (by  values  in  the  given  year) ;  and  II  and  ///  (by 
the  fictitious  values  found  by  multiplying  the  prices  of  one  year  by  the 
quantities  of  the  other).  And,  for  the  aggregative  type,  we  noted  two 
methods  of  weighting  index  numbers  of  prices  by  quantities,  viz.  I  (by 
quantities  in  the  base  year)  and  IV  (by  quantities  in  the  given  year). 

Finally,  in  Chapter  VIII,  we  used  weights  obtained  by  averaging  the 
weights  of  the  opposite  systems,  /  and  IV,  or  77  and  777.  These  weights 
were  usually  averaged  geometrically  but,  in  some  cases,  they  were  done 
arithmetically  and  harmonically  and  might  have  been  so  done  in  all. 

We  are  now  ready  to  answer,  with  some  precision,  the  question :  What 
differences  do  different  systems  of  weighting  make  in  the  resulting  index 
numbers?  We  have  already,  in  Chapter  V,  seen  that  a  biased  system  of 
weighting  makes  a  very  considerable  difference  in  the  index  number,  — 
substantially  the  same  difference  as  does  a  biased  type  of  index  number. 
Thus  (for  all  except  aggregatives)  weightings  777  and  IV  raise,  while  7 
and  77  depress,  any  index  number.  We  may  here,  for  convenience,  think 
of  this  effect  as  measured  relatively  to  a  cross  weight  index  number  which 
will  lie  about  midway  between  the  index  numbers  weighted  7  and  77,  on 
the  one  hand,  and  the  index  numbers  weighted  777  and  IV  on  the  other. 

439 


440         THE  MAKING  OF  INDEX  NUMBERS 

In  the  case  of  the  arithmetic,  geometric,  and  harmonic  index  numbers, 
the  upward  bias  of  weighting  ///  and  IV  and  the  downward  bias  of  /  and 
II  amounted,  in  our  example  of  36  commodities  for  1917  (on  1913  as  base), 
to  about  five  per  cent. 

The  reason  for  so  large  an  influence  of  weighting  was  the  bias  itself  — 


Simple  vs.  Cross-Weighted 
Index  Numbers 

(Prices) 


75 


'17 


78 


CHART  66P.  Showing  the  difference  which  different  weightings  make 
when  uncomplicated  by  bias.  The  differences  are  very  similar  in  the  cases 
of  the  arithmetic,  harmonic,  and  geometric,  but  not  very  similar  in  the  cases 
of  the  median,  mode,  and  aggregative. 


APPENDIX  II 


441 


the  fact,  for  instance,  that  by  the  weighting  system  IV  the  bigger  a  price 
relative  the  more  heavily  it  tends  to  be  weighted  and  the  smaller,  the  more 
lightly. 

But  if  we  take  systems  of  weighting  in  which  the  cards  are  not  thus 
stacked,  i.e.  systems  devoid  of  bias,  we  shall  find  that  differences  in  systems 
of  weighting,  —  even  very  wide  differences,  —  make  remarkably  small 
differences  to  the  resulting  index  numbers. 

The  failure  to  distinguish  between  the  effects  of  bias  in  the  weighting 


Simple    vs.  Cross-lighted 
Index  Numbers 

(Quantities) 


I** 


75  16  '17 

CHART  66Q.    Analogous  to  Chart  66P. 


442 


THE  MAKING  OF  INDEX  NUMBERS 


(which  are  important)  and  those  of  mere  blind  chance  (which  are  usually 
not  very  important)  is  responsible  for  much  of  the  confusion  on  this  sub- 
ject and  the  existence  of  two  apparently  opposite  opinions:  one,  that 
weighting  is,  and  the  other,  that  it  is  not,  important. 

,2 


Simple  vs.  Cross-Weighted 
Index  Numbers 
(Prices.Cont) 


/'I004 


Iff 


'0  74  IS  '/*  77  W 

CHART  67P.  Showing  the  differences  which  different  weightings  make 
when  uncomplicated  by  bias  to  the  factor  antitheses  of  the  index  numbers 
in  Chart  66P.  The  differences  correspond  to  those  in  Chart  66Q. 


APPENDIX  II  443 

§  2.  Simple  and  Cross  Weight  Index  Numbers  Compared 

The  two  unbiased  systems  of  weighting  which  have  been  set  forth  in 
this  book  are  simple  weighting  (the  weights  being  all  equal)  and  cross 
weighting  (the  weighting  being  averages  of  the  weights  under  systems 
7  and  IV,  or  //  and  ///). 

The  cross  weight  system  is  a  careful  and  discriminating  system  of  weight- 
ing, every  weight  taking  due  account  of  all  the  data  bearing  on  the  case  ; 
while  the  simple  is  a  careless  and  indiscriminate  system  which  shuts  its 
eyes  to  all  the  differences  among  commodities.  The  weights  in  the  two 
systems  —  cross  weight  and  simple  —  differ  enormously,  far  more,  in 
fact,  than  the  weights  of  /  and  IV t  or  of  //  and  ///. 

Simple  vs.  Cross-Weighted 
Index  Numbers* 

(Quantities.  Cont) 


•&  */4  75  16  '17  '/8 

CHART  67Q.    Analogous  to  Chart  67P.    The  differences  correspond  to 
those  in  Chart  66P. 


444         THE  MAKING  OF  INDEX  NUMBERS 

The  cross  weight  formula  for  the  arithmetic  is  1003  (not  1103),  and  for 
the  harmonic  1013.  The  other  cross  weight  formulae  are  1123,  1133,  1143, 
1153.  Thus  Formula  1003  is  a  weighted  arithmetic  index  number  freed 
of  weight  bias,  but  not  freed  of  the  (upward)  type  bias,  inherent  in  the  arith- 
metic type.  Likewise,  1013  is  a  weighted  harmonic,  freed  of  weight  bias 
but  not  freed  of  downward  type  bias. 

We  may  now  compare  each  of  these  six  weighted  index  numbers,  with 
the  corresponding  one  of  the  six  simple  or  evenly  weighted  index  numbers, 
all  twelve  being  free  of  weight  bias. 

Charts  66P  and  66Q,  67P  and  67Q  compare  the  simple  and  cross  weight 
index  numbers. 

§  3.  The  Differences  Haphazard 

The  first  point  which  strikes  us  in  these  comparisons  between  simple 
and  cross  weight  index  numbers  is  that  there  is  no  constant  tendency  for 
one  of  the  two  to  be  above  or  below  the  other.  The  two  curves  inter- 
twine, differing  either  way  and  about  equally  often.  It  is  a  matter  of  even 
chance,  not  of  bias. 

§  4.  The  Differences  among  the  Various  Similar  Types  of  Index  Numbers 

The  second  point  which  arrests  attention  is  the  remarkable  similarity 
in  the  influence  of  the  different  weighting  in  the  case  of  the  three  chief  types 
of  index  numbers.  That  is,  the  difference  between  the  simple  and  cross 
weight  arithmetics  is  practically  the  same  as  that  between  the  simple  and 
cross  weight  harmonics,  and  as  that  between  the  simple  and  cross  weight 
geometries. 

The  other  three  types  show  peculiarities,  though  not  always.  The  me- 
dians usually  behave  somewhat  similarly  to  the  first  three,  the  arithmetic, 
harmonic,  and  geometric.  But  the  modes  are  erratic  compared  with  the 
first  three  and  with  each  other.  The  simple  aggregative  is  very  erratic, 
while  the  cross  weight  aggregative  is  not. 

§  6.  The  Differences  Small 

The  third  point  which  strikes  us  in  making  these  comparisons  is  how 
surprisingly  small  is  the  difference  made  by  using  the  careful  discriminating 
cross  weighting  instead  of  the  erratic  simple  weighting.  This  is  aston- 
ishing when  we  consider  that  the  two  sets  of  weights  themselves  differ 
enormously.  In  the  simple  weighting  all  36  commodities  are  equally 
important  while  in  the  cross  weighting  (in  the  case  of  the  price  index 
number  in  1914,  for  instance)  the  highest  weight  (that  for  lumber)  was 
118  times  as  great  as  the  lowest  (that  for  skins) ;  in  1915  the  highest  was 
134  times  the  lowest;  in  1916,  it  was  100  times;  in  1917,  130  times;  and 
in  1918,  261  times.  Yet,  in  spite  of  these  enormous  variations  (and  in 
spite  of  the  fact  that  there  are  only  36  commodities  in  the  list),  these 
unbiased  (simple  and  cross  weighted)  forms  usually  agree  within  five  or 
ten  per  cent.  In  fact,  out  of  60  comparisons  between  the  simples  and 
cross  weighted  index  numbers  (for  both  prices  and  quantities),  there  are 


APPENDIX  II  445 

only  13  differences  exceeding  five  per  cent  and  only  five  over  ten  per  cent. 
In  the  case  of  the  arithmetic,  harmonic,  and  geometric,  there  is  only  one 
instance  of  a  discrepancy  over  eight  per  cent.  This  is  for  1918  for  the 
harmonic  where  there  is  a  discrepancy  of  over  30  per  cent. 

The  reason  for  this  large  discrepancy  is  to  be  found  in  one  commodity, 
skins,  the  quantity  of  which  fell  between  1913  and  1918  tenfold.  Although 
this  enormous  fall  is  quite  out  of  tune  with  the  general  movement  of  the 
other  35  commodities,  nevertheless  it  ought  properly  not  to  have  much 
influence  on  the  average  change  of  the  36  commodities  because  "skins" 
was  so  insignificant  a  commodity.  And  in  the  weighted  average  this  is  the 
case,  since  " skins"  is  given  only  ^rtnr  of  the  total  weight.  But  by  the 
simple  weighting  its  influence  is  ^  of  the  total  which  is  nearly  a  hundred 
times  the  influence  it  should  have. 

Such  a  great  change,  as  in  the  quantity  of  skins,  is  almost  never  met  with 
and  when  it  does  occur  it  is  usually  smothered  up  by  the  other  commodi- 
ties because  of  there  being  so  many.  In  fact,  it  is  smothered  even  in  the 
present  case  of  only  36  commodities,  except  where  the  harmonic  method  is 
used,  which  gives  a  special  emphasis,  as  it  were,  to  terms  unusually  small. 
Probably  such  a  freak  effect  would  not  be  encountered  once  in  a  hundred 
times  in  the  ordinary  course  of  using  index  numbers. 

Professor  Wesley  C.  Mitchell  cites  many  actual  examples  l  of  the  effect 
of  weighting  as  compared  to  simple  index  numbers.  In  general,  the  differ- 
ences are  less  even  than  those  here  found,  being  seldom  ten  per  cent,  except 
under  the  chaotic  conditions  created  by  the  greenback  standard  in  1862- 
1878.  Ordinarily  the  difference  between  the  simple  and  the  best  weighted 
index  number  of  the  Aldrich  Senate  Report  was  less  than  three  per  cent. 

The  influence  of  a  change  of  weighting  is,  of  course,  different  for  dif- 
ferent types  of  index  numbers.  In  general,  a  given  change  in  weighting 
produces  least  effect  in  the  mode,  somewhat  more  in  the  median,  and  very 
much  more  in  the  arithmetic,  harmonic,  and  geometric.  For  the  aggrega- 
tive formula  the  process  of  weighting  has  a  different  meaning  from  what 
it  has  for  the  rest,  being  a  matter  of  quantities  only.  The  effect  of  a  change 
in  these  quantities  on  the  index  number  is  small.  It  is  about  the  same  as 
the  effect  of  a  change  in  the  weighting  of  the  arithmetic,  harmonic,  and 
geometric,  when  only  quantities  are  changed.  Thus  there  is  very  little 
difference  between  the  aggregatives,  53  and  59,  dependent  on  a  change  in 
quantities  only,  —  about  the  same  difference  as  between  the  arithmetics 
3  and  5,  or  7  and  9,  or  the  harmonics  13  and  15,  or  17  and  19,  geometries 
23  and  25,  or  27  and  29,  in  all  of  which  cases  the  only  change  is  in  quan- 
tities. 

The  effect  of  a  change  in  weights  is  more  spasmodic  or  irregular  in  the 
cases  of  the  mode  and  median  than  in  that  of  the  other  four  types.  This 
is  true  even  when  bias  is  involved.  Thus  there  is  no  appreciable  difference 
between  the  modes,  43  and  49,  and  little  between  the  medians,  33  and  39, 
—  and  that  little,  spasmodic.  There  is  much  more  difference  between 
the  arithmetics,  3  and  9,  or  the  harmonics,  13  and  19,  or  the  geometries 
23  and  29. 

1  Bulletin  284,  United  States  Bureau  of  Labor  Statistics,  pp.  61-62. 


446 


THE  MAKING  OF  INDEX  NUMBERS 


§  6.  Bias  More  Disturbing  than  Chance 

We  have  seen  that  in  the  case  of  the  two  unbiased  weighting  systems, 
the  simple  and  the  cross,  while  the  weights  often  vary  a  hundredfold,  the 
resulting  index  numbers  seldom  differ  over  five  per  cent.  But  the  biased 
forms,  7  and  IV  for  instance,  differ  often  eight  or  ten  per  cent,  although 
the  weights  never  differ  even  as  much  as  twofold. 

This  conclusion,  that  bias,  even  when  weights  vary  little,  is  more  dis- 
turbing than  chance,  even  when  weights  vary  enormously,  may  be  still 
more  definitely  illustrated.  If  we  take  the  36  commodities  at  random, 
i.e.  regardless  of  their  importance  as  to  p's  or  g's  —  let  us  say,  alphabeti- 
cally,—  and  divide  them  into  two  groups  of  18  each,  and  then  multiply  the 
weights  (quantities)  of  the  first  group  by  ten,  the  index  number  for  1917 
(the  year  most  likely  to  create  a  disturbance)  becomes,  by  Formula  53, 
175.20  per  cent  instead  of  162.07,  a  difference  of  8  per  cent.  Now  observe 
what  happens  if,  instead  of  taking  our  two  lists  of  18  at  random,  we  select 
them  so  that  the  first  18  will  be  those  which  will  have  the  greatest  influence 
in  raising  the  result.  When  these  hand  picked  18  are  increased  tenfold 
in  importance  the  result  is  201.33,  exceeding  162.07  by  24  per  cent.  The 
contrast  in  effects  is  shown  in  Table  59 : 

TABLE  59.  COMPARATIVE  EFFECTS  ON  THE  INDEX  NUM- 
BER FOR  1917  (BY  FORMULA  3)  OF  INCREASING  TEN- 
FOLD THE  WEIGHTS  OF  HALF  OF  THE  36  COMMODITIES 
ACCORDING  AS  THE  COMMODITIES  ARE  TAKEN  AT  RAN- 
DOM, OR  SELECTED  TO  MAKE  THE  LARGEST  EFFECT 


INDEX  NUMBER 

By  using  the  true  weights          . 

162.07 

By  falsifying  18  weights  tenfold  at  random     .... 
By  falsifying  18  weights  tenfold  by  selection    .... 

175.20 
201.33 

Let  us  note  the  small  effect  in  the  index  number  of  increasing  tenfold 
the  weight  (quantity)  of  skins,  the  commodity  which  shows  the  greatest 
aberrations  from  the  general  course  of  prices  of  the  36  commodities.  Tak- 
ing Formula  1153,  for  instance,  we  find  the  following  effects  on  the  index 
numbers  of  prices  on  1913  as  base : 

TABLE  60.   INDEX  NUMBERS  COMPUTED  BY  USING 
DIFFERENT  WEIGHTS  FOR  SKINS 


WEIGHT  USED 

1913 

1914 

1915 

1916 

1917 

1918 

True     

100 

100.13 

99.89 

114.20 

161.70 

177.83 

Ten  times  true  .     .     . 

100 

100.15 

99.93 

114.66 

162.05 

177.96 

APPENDIX  II 


447 


The  effect  of  even  this  enormous  increase  of  the  weight  of  the  most  erratic 
commodity  is  negligible. 

In  this  case  the  effect  was  small  because  skins  had  originally  so  small  a 
weight.  I  have,  therefore,  tried  to  find  the  commodity  in  whose  case  in- 
creasing the  weight  would  most  affect  the  index  number.  This  seems  to  be 
hay  which,  though  not  as  erratic  as  skins,  has  much  more  weight  to  start 
with.  We  find,  using  the  same  formula,  the  following  results : 

TABLE  61.    INDEX  NUMBERS  COMPUTED  BY  USING 
DIFFERENT  WEIGHTS  FOR  HAY 


WEIGHT  USED 

1913 

[11914 

1915 

1916 

1917 

1918 

True  

100 

100.13 

99.89 

114.20 

161.70 

177.83 

Ten  times  true      .    . 

100 

103.75 

101.27 

104.29 

159.71 

184.04 

This  case  is  extreme  (1)  because  the  commodity  is  extreme,  being  chosen 
for  its  big  influence,  (2)  because  its  influence  is  magnified  by  the  fact  of 
there  being  only  36  commodities,  and  (3)  because  the  change  in  the  weight 
(tenfold)  is  extreme.  Yet  even  under  all  these  circumstances  combined 
the  effect  of  the  change  in  weight  does  not  exceed  3.6  per  cent  except  in 
one  instance  when  it  reaches  nearly  ten  per  cent. 

Such  hand  picked  instances  as  those  just  described  are  not,  of  course, 
fair  or  representative  of  the  actual  situations  with  which  the  computer  of 
index  numbers  has  to  deal.  Ordinarily  inaccuracy  in  weights  will  not  pro- 
duce appreciable  effects  because  (1)  any  inaccuracy  is  not  likely  to  be  very 
great,  such  as  100  per  cent,  much  less  tenfold ;  (2)  if  it  does  happen  to  be 
great  it  is  not  likely  that,  at  the  same  time,  the  commodity  to  which  it 
attaches  will  be  very  important  or  very  erratic,  much  less  both  important 
and  erratic ;  (3)  if  some  of  these  things  do  conspire,  there  is  still  a  good 
chance  that  opposite  errors  elsewhere  will  largely  offset  the  effect ;  (4)  even 
at  the  worst  the  effect  is  greatly  reduced  if  a  large  number  of  commodities 
is  used ;  the  average  commodity  in  a  list  of  100  commodities  might  deviate 
from  the  general  average  100  per  cent  without  affecting  the  final  result 
by  one  per  cent. 


§  7.  Errors  in  Weights  Less  Important  than  in  Prices 

Correct  weights  in  an  index  number  of  prices  are  far  less  important  than 
correct  prices.  Chart  68  shows  the  index  numbers  by  Formula  3  for  1914 
and  1917,  and  shows  (1)  what  it  becomes  if  the  weight  of  any  one  of  the 
36  commodities  is  doubled,  the  weights  of  the  rest  being  unchanged,  as 
well  as  (2)  what  it  becomes  if  the  price,  relatively  to  1913,  of  any  one  is 
doubled  (the  prices  of  the  rest  being  unchanged). 

It  will  be  noted  that  the  doubling  of  the  weight  does  not  greatly  swerve 
the  1914  figure  from  the  original  99.93.  The  largest  increase  is  produced 
by  doubling  the  weight  of  hay  which  raises  the  index  number  from  this 


448         THE  MAKING  OF  INDEX  NUMBERS 

99.93  to  100.54,  or  about  half  of  one  per  cent,  and  the  largest  decrease 
is  produced  by  doubling  the  weight  of  bituminous  coal  or  pig  iron 
which  lowers  the  index  number  to  99.59.  Doubling  the  price,  on  the 
other  hand,  changes  the  figure  very  considerably,  causing  it  to  reach  115.07 
when  lumber  is  doubled  in  price. 

The  same  contrast  is  exhibited  in  1917.  Doubling  a  weight  changes 
162.07  at  most  to  167.36  (in  the  case  of  bituminous  coal)  while  doubling  a 
price  raises  the  162.07  to  179.54  in  the  case  of  lumber.  The  average 

Weighting  is  Relatively  Unimportant 

igtf  Effect  of  Doubling  a  Price 

Cffect  of  Doubling  a  Weight 

\S% 
Effect  of  Doubling  a  Price 

1914  "          A/A 

Effect  of  Doubling  a  Weight  In^m 


CHART  68.  Showing  that  if  the  weighting  of  (say)  barley  is  doubled, 
the  index  number  for  1914  is  slightly  decreased  and  that  for  1917  is  slightly 
increased,  while  if  the  price  relative  of  barley  is  doubled  the  index  number 
is  greatly  increased  in  both  cases. 

change  produced  by  doubling  the  weight  is  .15  for  1914  and  1.08  for  1917, 
while  the  average  change  produced  by  doubling  the  price  is  2.77  for  1914 
and  4.49  for  1917.  Reduced  to  percentages  of  the  index  numbers  them- 
selves, doubling  a  weight  affects  it  on  the  average  .15  per  cent  in  1914  and 
.67  per  cent  in  1917  and  doubling  a  price  affects  it  2.78  per  cent  in  1914 
and  2.77  per  cent  in  1917. 

Thus  the  effect  produced  by  doubling  a  price  is,  in  1914,  18  times  the 
effect  produced  by  doubling  a  weight,  and,  in  1917,  four  times.  These 
figures  measure  the  relative  importance  of  accuracy  in  prices  and  accuracy 
in  weights.  The  latter  is  comparatively  unimportant.  Rough  estimates 
and  even  guesses  in  selecting  weights  are  admissible  but  guess  work  in 
selecting  price  data  is  dangerous.  However,  weighting  increases  in  im- 

1  The  simple  arithmetic  average  change  from  the  original  figure  disregarding  direction 
of  change. 


APPENDIX  II  449 

portance  with  an  increase  in  the  dispersion  of  prices.  In  1914,  when  the 
dispersion  was  small,  doubling  a  weight  had  less  effect  than  in  1917  when 
the  dispersion  was  larger.  A  formula  could  be  worked  out  connecting 
dispersion  with  the  effect  of  weighting,  but  it  would  be  different  for  different 
sorts  of  index  numbers. 

These  results  are  representative.  But  it  should  be  noted  that,  in  ex- 
ceptional cases,  doubling  the  weight  may  produce  an  effect  equal  to,  or 
greater  than,  doubling  the  price.  Thus,  if  an  individual  price  relative  is 
almost  zero  (say  one  per  cent)  while  the  average  of  all  is  high  (say  100 
per  cent),  doubling  the  price  relative  from  one  to  two  per  cent  will  evi- 
dently produce  only  an  infinitesimal  effect  on  the  average  100  —  a  mere 
fraction  of  one  per  cent  —  while  doubling  the  weight,  if  the  commodity 
already  has  a  heavy  weight,  will  pull  the  index  number  down  a  con- 
siderable part  of  the  98  per  cent  deviation  between  that  commodity's 
low  price  relative  and  the  high  original  average,  100.  In  practice,  how- 
ever, such  cases  are  rarely,  if  ever,  met  with. 

§8.  What  Weights  are  Best?; 

In  view  of  what  has  been  said  it  is  clear  that  weights  may  be  at  fault 
either  because  they  are  erratic  or  because  they  have  a  wrong  bias.  As 
to  the  former,  all  will  agree  that  "simple"  weighting,  being  usually  very 
erratic,  should  be  avoided  whenever  possible.  As  to  bias,  the  matter  is 
not  so  simple.  We  must  not  jump  to  the  conclusion  that  cross  weights 
are  always  best.  They  are  best  for  the  geometric,  median,  mode,  and 
aggregative ;  but  for  the  arithmetic,  the  best  weighting  is  the  biased  weight- 
ing 7  or  II,  and  for  the  harmonic  the  best  weighting  is  the  biased  weight- 
ing 777  or  IV,  because  the  upward  bias  possessed  by  the  arithmetic  type 
needs  to  be  counteracted  by  a  downward  bias  in  the  weighting,  and  the 
downward  bias  of  the  harmonic  needs  an  upward  bias  in  the  weighting. 

It  has  usually  been  assumed  that  the  problem  of  finding  the  best  formula 
for  an  index  number  consists  of  two  separate  problems :  (1)  to  find  the  best 
type,  and  (2)  to  find  the  best  weighting.  But  these  two  problems  cannot 
be  separated,  for  the  weight  which  is  best  for  one  type  is  not  best  for  an- 
other. 

What  has  been  said  applies  to  the  primary  formulae.  The  system  of 
weighting  immediately  sinks  into  insignificance  when  we  cross  these  for- 
mulae to  rectify  them.  Even  such  an  absurdly  weighted  formula  as  9, 
where  upward  biased  weights  are  applied  to  exaggerate  the  already  upward 
biased  type,  the  arithmetic,  when  rectified  by  crossing  with  the  like  doubly 
biased  harmonic,  13,  yields  an  excellent  and  unbiased  result,  109.  In 
short,  rectification  will  cure  bad  weighting  if  the  badness  is  systematically 
biased. 

But  if  the  fault  is  merely  that  the  weighting  is  erratic,  as  in  the  case  of 
the  simple  index  number,  the  rectification  by  Test  1  will  be  of  little  avail. 
Rectification  by  Test  2  will  help  more,  but  not  completely.  In  short,  bias 
can  be  neutralized  by  bias,  but  freakishness  is  nearly  incorrigible. 

Thus  the  simple  Formulae  1,  11,  21,  31,  41,  51  are  freakishly  weighted. 
Crossing  Formulae  1  and  11  gives  101  which  is  practically  identical  with 


450         THE  MAKING  OF  INDEX  NUMBERS 

21.  Thus  Formula  101  as  well  as  21,  31,  41,  51  are  free  from  bias  but  not 
from  freakishness.  Crossing  each  with  the  next  following  even  numbered 
formulae  (viz.  102,  22,  32,  42,  52),  their  factor  antitheses,  we  get  301,  321, 
331,  341,  351,  which  are  only  slight  improvements  over  the  originals. 

§  9.  Summary 

We  may  summarize  the  main  points  in  this  Appendix  as  follows : 

(1)  The  greater  the  number  of  commodities  in  an  index  number  of 
prices  the  less  is  the  index  number  affected  by  a  change  in  weights,  or  in 
price  relatives. 

(2)  A  change  in  a  weight  has  far  less  influence  than  a  change  in  a  price 
relative. 

(3)  The  contrast  between  two  index  numbers  having  weights  of  oppo- 
site bias  is  greater  than  that  between  the  simple  and  the  cross  weight  index 
numbers,  in  spite  of  the  fact  that  the  variations  in  the  size  of  the  weights 
are  immensely  greater  in  the  latter. 

(4)  A  biased  type  of  formula  may  be  largely  corrected  by  using  an 
oppositely  biased  sort  of  weighting. 

(5)  Bias  disappears  by  rectification.    Freakishness  does  not. 


APPENDIX  III 

AN  INDEX  NUMBER  AN  AVERAGE  OF  RATIOS  RATHER 
THAN  A  RATIO  OF  AVERAGES 


§  1.  Introduction 

An  index  number  should  be  an  average  of  ratios  rather  than  a  ratio  of 
averages.  There  are  always  these  two  ways  of  averaging  the  data  from 
which  index  numbers  are  constructed.  Thus,  for  36  commodities,  we  may 
either  (1)  average  the  36  figures  for  one  of  the  two  years  taken  by  itself 
and  again  average  the  36  figures  for  the  other  year  taken  by  itself,  and  then 
obtain  the  ratio  between  these  two  averages,  or  (2)  we  may  take  each  in- 
dividual commodity  and  calculate  its  own  special  ratio,  or  relative,  as  be- 
tween the  two  years  and  then  average  these  36  relatives.  The  first  way 
is  to  take  one  ratio  of  two  averages;  the  second  is  to  take  one  average  of  36 
ratios. 

As  applied  to  prices,  the  first  method  tells  us  the  change  in  the  average  of 
various  prices  of  commodities ;  the  second  tells  us  the  average  of  the  various 
changes  of  prices.  These  two,  though  usually  confused,  are  very  distinct. 
The  latter  is  much  the  more  essential  concept ;  the  former,  though  it  can 
be  computed,  is  apt,  in  general,  to  prove  a  delusion  and  a  snare.  The 
reason  is  that  an  average  of  the  prices  of  wheat,  coal,  cloth,  lumber,  etc., 
is  an  average  of  incommensurables  and  therefore  has  no  fixed  numerical 
value.  It  can  be  calculated,  but  the  resulting  figure  depends  arbitrarily 
on  the  units  we  happen  to  choose.  The  index  number  is  thus  indetermi- 
nate, yielding  different  results  for  every  different  kind  of  measure.  If 
wheat  is  $1  a  bushel,  coal  $10  a  ton,  cloth  $2  a  yard,  and  lumber  $20  a 

thousand  board  feet  we  may  say  that  l  +  10  +  2  +  2°  =  $8.25  is  the 

average  price  of  these  four  commodities  "per  unit."  Suppose  the  four 
prices  above  mentioned  to  be  the  prices  for  1913  and  suppose  the  four  prices 
in  1918  to  be  different,  as  per  the  following  table : 


1913 

1918 

$   1 

$  2 

10 

10 

2 

3 

Lumber  per  thousand 

20 

50 

$  825 

$16  25 

451 


452         THE  MAKING  OF  INDEX  NUMBERS 

The  index  number,  as  the  ratio  of  these  average  prices,  is      '     or  197  per 


cent.  But  as  there  are  four  entirely  separate  and  incommensurable  units, 
any  one  of  which  can  be  changed  without  entailing  change  in  the  others, 
it  is  clear  that  this  "average  price"  is  really  an  unstable  compound.  If 
we  choose  to  have  coal  measured  by  the  hundredweight  its  price  must  be 
regarded  no  longer  as  $10  but  as  50  cents,  and  this  without  requiring  any 
corresponding  change  in  the  prices  of  wheat,  cloth,  or  lumber.  The  "aver- 
age price"  then  for  1913  becomes  *  +  -50  +  2  +  20  =  $5  g?  per  <<unit  » 

4 

13  87 
The  "average  price"  for  1918  becomes  $13.87  giving  —  —  ,  or  236  per  cent 

5.87 
as  the  index  number. 

Thus,  simply  by  changing  at  will  the  unit  of  measuring  coal,  even  though 
it  is  changed  in  both  numerator  and  denominator,  we  change  the  index 
number  from  197  to  236  ! 

When  this  method  is  applied  to  the  case  of  the  36  commodities,  their 
average  price  in  1913  is  found  to  be  6.636  and  in  1918,  11.464,  the  ratio  of 
which  is  172.76  per  cent  as  compared  with  the  "  ideal." 

The  case  above  mentioned  is  really  Formula  51  in  our  table.     For  the 
formula  for  the  average  price  of  commodities  in  year  "0"  is  evidently 
Po  +  p'o  +  p"o  +  .  .  .  or  2po  where  n  ig  the  number  of  commodities,  while 
n  n 

the  corresponding  average  price  for  year  "  1  "  is  —  —  ,  making  the  index 

n 

number  for  year  "1"  relatively  to  year  "0" 


But,  cancelling  n,  this  becomes  — ^,  our  Formula  51.    This  cancellation 

2p0 

assumes,  of  course,  that  the  number  of  commodities  averaged  is  the  same 
in  both  years. 

Formula  51  and  its  derivatives  52,  151,  152,  251,  351  are  the  only  for- 
mulae in  our  list  which  have  the  incommensurable  defect  due  to  taking  a 
ratio  of  averages  and  are  affected  by  a  change  in  units  of  measurement. 
I  have  included  them,  however,  partly  because  51  is  actually  used  by  Brad- 
street  and  partly  because  51  seemed,  so  far  as  any  formula  can  be  said  to 
do  so,  to  fill  in  the  otherwise  vacant  space  for  a  "simple  aggregative." 

§  2.  Some  Ratios  of  Simple  Averages  Calculated 

For  the  reader  who  is  curious  to  see  what  the  corresponding  ratio  of  aver- 
ages would  be  like  for  the  various  types  the  following  notes  are  added.  I 
have  gone  through  the  calculations  because  I  find  even  experienced  workers 


APPENDIX  III  453 

in  index  numbers  are  confused  on  this  subject  and  do  not  seem  to  realize 
that  the  ratio  of  average  method  is  untrustworthy. 

The  simple  harmonic  average  of  prices  is       -^—  for  year  "0"  and 


for  year  "1."    The  index  number  in  the  sense  of  the  ratio  of  these 


*(± 

\PI> 
averages  is,  therefore, 


This  formula,  like  51,  could  be  used  if  the  units  of  measure  were  judi- 
ciously selected,  but  there  would  be  no  object  in  using  it.  For  our  36 
commodities  it  gives  as  the  ratio  of  the  simple  harmonic  average  of  the 
prices  for  1918  to  the  corresponding  average  for  1913,  165.67  per  cent. 

The  simple  geometric  average  of  prices  is  -y/po  X  p'o  X  p"o  X  .  .  . 
for  year  "0"  and  the  corresponding  formula  for  year  "1."  The  index 
number  is  their  ratio.  Evidently  this  can  be  reduced  to  Formula  21  and 
is  the  average  of  ratios.  It  gives  the  index  number  of  prices  for  1918  rela- 
tively to  1913  as  180.12.  In  the  case,  therefore,  of  the  simple  geometric 
we  get  the  same  result  whether  we  take  the  ratio  of  averages  or  the  average 
of  ratios,  assuming  the  same  number  (n)  in  both  years. 

The  simple  median  and  simple  mode  of  prices  are  even  more  absurd 
than  the  simple  arithmetic  and  the  simple  harmonic.  Thus,  for  the 
median,  after  arranging  in  the  order  of  magnitude  the  prices  of  1913  and 
those  of  1918,  we  find  that  the  median  price  in  1913  lies  between  the  price 
of  barley,  which  is  .6263  per  bushel,  and  rubber  which  is  .8071  per  pound, 
and  maybe  taken  as  their  (geometric)  mean,  .7110,  while  the  median  price  in 
1918  lies  between  the  price  of  barley,  which  is  1.4611  per  bushel,  and  wool, 
which  is  1.66  per  pound,  and  may  be  taken  as  their  (geometric)  mean, 
1.5574.  The  ratio  of  these  two  medians  is  219.05  per  cent,  an  absurd  result. 

There  remains  only  the  aggregative  method.  This  method  is  scarcely 
applicable  for  taking  an  average  of  prices  or  of  quantities.  It  is  certainly 
not  applicable  at  all  to  averaging  quantities  since  a  quantity  is  not  a  ratio, 
and  the  aggregative  method  of  averaging  implies  ratios,  the  numerators 
of  which  are  to  be  added  together  and  the  denominators  likewise.  As  to 
prices,  if  we  choose  to  go  back  of  the  individual  price,  each  price  is  resolv- 
able into  a  ratio  of  a  quantity  of  money  to  a  quantity  of  a  commodity  sold 
for  that  money  and  we  can,  of  course,  add  together  the  money  spent  by  a 
specified  group  of  people  on  all  the  commodities  for  the  numerator  and,  for 
the  denominator,  add  together  the  number  of  bushels,  tons,  yards,  board 
feet,  etc.  But  this  procedure  would  be  as  impracticable  as  it  would  be  use- 


454         THE  MAKING  OF  INDEX  NUMBERS 

less  and  arbitrary.  The  result  would  be  the  same  as  that  for  the  weighted 
arithmetical  average  price  method  which  follows  next. 

§  3.   Some  Ratios  of  Weighted  Averages  Calculated 

The  weighted  arithmetic  average  of  prices,  if  the  weights  be  the  quanti- 
ties, gives     ^°p°  as  the  average  price  per  "unit  in  1913."   The  numerator  of 

Sgo 

this  fraction  is,  it  is  true,  homogeneous ;  it  is  not  a  sum  of  incommensur- 
ables  but  a  sum  of  money  values.  The  denominator,  however,  is  made  up 
of  incommensurables.  Consequently,  the  resulting  average  itself  is  de- 
pendent for  its  particular  numerical  value  on  the  accident  of  what  par- 
ticular units  of  measurement  happen  to  be  employed. 

The  "  index  number  "  for  1914  relative  to  1913  then  becomes 


In  this,  the  S^i  and  Sg0  are,  neither  of  them,  homogeneous  and,  what 
is  here  the  vital  point,  they  are  not  equal  and  so  do  not  cancel  out.  Con- 
sequently they  vitiate  the  resulting  index  number  which  is  likewise  de- 
pendent on  the  particular  units  chosen  and,  therefore,  absurd  as  an  index 
number.  It  reduces  to 


which  is  Formula  52  in  our  series,  but  one  of  the  worst. 

Let  us  now  calculate  the  index  number  which  the  last  formula  represents 
—  the  ratio  of  the  weighted  arithmetic  average  of  prices  in  1918  to  1913 
for  our  36  commodities.  Taking  the  price  quotations  as  they  stand  we 
find  the  arithmetic  average  of  the  prices 

forl918  is  29186.105 


and  for  1913  is         °  =  13104'818  =  .308861 
42429.44 

The  index  number  is,  therefore,  the  former  divided  by  the  latter,  or  165.14 
per  cent.  But  this  index  number  is  built  on  quicksands.  For  no  one 
could  complain  if  in  our  reckoning  the  quotation  of  cotton  was  made  per 
bale  instead  of  per  pound.  To  take  an  extreme  illustration  which  will 
show  in  an  extreme  degree  the  absurdity  of  the  results  obtained  by  this 
formula  by  simply  changing  the  unit  of  measurement,  let  us  measure  rubber 
in  grains  instead  of  in  pounds.  Under  these  circumstances 

29186.105 


2517368.25 


APPENDIX  III  455 

13104.818  m  Q15365 
S?o          852913.64 

and  the  index  number  is  75.46  per  cent'. 

Which  shall  we  choose,  the  165.14  or  the  75.46?  Evidently  an  index 
number  so  constructed  would  be  indeterminate  unless,  as  a  part  of  its  speci- 
fications, we  prescribe  every  unit  of  measure  to  be  used  in  its  calculation ! 

But  if  we  alter  the  numerator  by  substituting  q0,  q'0,  etc.,  for  qi}  q'if 
etc.,  the  formula  becomes 


Stfo 
in  which  the  Sg0  may  be  canceled  leaving 


or  Formula  53. 

Or,  we  may  alter  the  denominator  by  substituting  gb  g'i,  etc.,  for  q0,  q'0, 
etc.,  in  which  case,  after  cancellation,  we  obtain  Formula  54.  In  both 
these  cases  the  cancellation  removes  all  traces  of  incommensurables. 

It  thus  turns  out  that  the  best  of  our  primary  formulae  (viz.  53  and  54) 
may  be  regarded  as  ratios  of  price  averages  which,  although  they  seem  at 
first  to  have  the  "  incommensurable"  defect,  are  really  free  of  it;  for  the 
incommensurables  are  the  same  in  numerator  and  denominator  and  so 
disappear  in  the  final  result.  And  such  an  index  number  as  53  or  54  is 
not  really  a  ratio  of  averages  of  prices  of  the  two  years.  Only  one  of  the 
two  figures  (the  denominator  for  Formula  53,  for  instance)  can  be  claimed 
as  the  true  average  price  of  the  year  referred  to.  The  other  had  to  be 
altered  in  order  to  insure  ultimate  cancellation  of  the  incommensurables. 
If  the  method  of  averaging  the  prices  were  a  good  one  in  this  case,  it  ought 
to  stand  on  its  own  feet  for  both  years. 

Let  us  next  take  the  geometric.  If  the  weights  be  p<#o,  p'<#'o,  etc.,  for 
year  "0"  and  p^i,  p\q'i,  etc.,  for  year  "1,"  the  ratio  of  the  geometric  aver- 
ages of  prices  is 


Taking  as  the  units  of  commodities  those  quoted  in  the  market,  this 
formula  gives  the  index  number  for  1918  relatively  to  1913  as  124.53  per 
cent.  But  if  we  change  lumber  from  M  board  feet  to  board  feet  the  same 
formula  gives  71.14  per  cent  !  Like  all  the  others,  therefore,  the  geometric 
ratio  of  averages  has  the  fatal  blight  of  incommensurability.  To  be  freed 
of  it,  it  is  necessary  to  alter  the  pq's  in  either  the  numerator  or  the  denomi- 
nator, or  both,  so  as  to  make  the  two  agree.  In  this  way  we  can  make 


456         THE  MAKING  OF  INDEX  NUMBERS 

the  method  of  averaging  prices  yield  results  given  by  the  other  method, 
that  of  averaging  ratios,  and  get  the  Formulae  23,  25,  27,  29,  and  6023. 

Thus  we  find  only  two  cases  where  this  defect  of  incommensurability 
disappears,  namely,  (1)  in  the  geometric  average  of  prices  with  constant  l 
weights,  the  ratio  of  which  yields  our  Formulae  21,  23,  25,  27,  29,  6023, 
and  (2)  the  arithmetic  average  of  prices  weighted  by  quantities  (provided, 
however,  these  quantities  are  taken  as  the  same  in  both  years)  which  yields 
our  53,  54,  and  6053. 

All  these  derive  their  immunity  from  the  incommensurable  taint  from 
the  fact  that  the  incommensurable  elements  cancel  out,  so  that  they  can 
be  reduced  to  an  average  of  price  ratios.  Moreover,  all  except  the  ratio  of 
the  simple  geometric  averages  (which  reduces  to  Formula  21)  have  to  be 
altered  before  they  can  be  reduced  to  an  average  of  ratios  and  even  the 
exception  named  presupposes  the  choice  of  the  same  number  of  commodities 
in  the  years  compared. 

In  short,  all  true  index  numbers  are  averages  of  ratios.  A  ratio  of 
averages,  unless  reducible  to  an  average  of  ratios,  is  subject  to  a  haphazard 
change  from  every  change  of  unit.  In  other  words,  it  fails  in  the  "com- 
mensurability  test "  (Appendix  I,  Note  to  Chapter  XIII,  §  9),  the  elemen- 
tary requirement  of  every  application  of  mathematics,  namely,  of 
possessing  homogeneity. 

§  4.  Cases  Where  Averages  of  Prices  can  Properly  be  Used 

The  only  cases  in  which  it  is  really  justifiable  to  use  the  genuine  method 
of  taking  the  ratio  of  averages  is  where  the  units  are  really  or  nearly  com- 
mensurable. Thus,  it  is  entirely  legitimate  to  obtain  the  index  number  of 
various  quotations  of  one  special  kind  of  commodity,  such  as  salt,  by  taking 
the  average  of  its  prices  in  different  markets.  In  such  a  case  the  precau- 
tion, so  essential  in  the  previous  examples,  of  forcibly  altering  numerator 
to  suit  denominator,  or  vice  versa,  does  not  need  to  be  taken.  The  true 
average  for  each  year  can  be  taken  independently  of  the  other  years.  An- 
other case  is  where  the  commodities  are  of  one  general  group,  such  as  kinds 
of  coffee  or  fuels,  e.g.  coal  and  coke  where  the  same  unit,  such  as  the  ton, 
is  used  for  all  so  that  there  is  no  danger  of  changing  one  without,  at  the 
same  time,  changing  the  others  equally. 

The  most  interesting  practical  examples,  however,  are  the  average  wage 
of  different  but  similar  kinds  of  labor  and  the  average  price  of  different 
but  similar  kinds  of  securities,  in  which  cases  the  objection  of  incommen- 
surability applies  but  not  very  strongly.  In  the  stock  market  the  aver- 
age price  of  stocks  is  taken,  the  "common  unit,"  if  it  may  be  so  called, 
being  the  "par  value." 

§  5.  Conclusion 

It  perhaps  does  not  greatly  matter  if  the  general  public  thinks  of  a  "price 
level"  as  something  which  can  be  calculated  for  each  year  independently 
of  other  years  and,  to  suit  this  concept,  it  is  possible  by  making  prices  in 

1  Constant  as  between  the  two  years  in  the  index  number,  not  necessarily  as  to  a  series 
of  years. 


APPENDIX  III  457 

"dollars*  worth"  of  one  year,  instead  of  in  pounds,  yards,  etc.,  to  expound 
the  subject  in  such  terms  before  an  elementary  class.  But  such  a  trans- 
position of  units  covertly  introduces  price  ratios.  The  method  of  taking 
the  ratio  of  average  prices  is  too  lame  to  walk  alone  and  needs  always  to 
lean  on  the  other  and  fully  trustworthy  method  of  averaging  the  price 
ratios. 

We  conclude,  then,  that  while  it  is  possible  to  calculate  an  index  number 
by  first  averaging  prices  for  the  two  years  and  then  taking  the  ratio  of  the 
two  averages,  this  procedure  has  one  of  two  faults.  Either  it  makes  the 
resulting  index  number  dependent  on  the  arbitrary  choice  of  units  of 
measure,  so  creating  "haphazard  weighting,"  or  it  requires  us  to  force  or 
falsify  one  of  the  two  averages  to  make  it  match  the  other  in  order  to 
enable  us  to  cancel  out  the  "incommensurable"  items;  in  the  latter  case, 
the  resultant  formula  turns  out,  after  all,  to  be  an  average  of  ratios.  In 
short,  the  ratio  of  averages  has  either  the  fault  of  being  haphazard  or 
the  fault  of  being  superfluous. 


APPENDIX  IV 

LANDMARKS  IN  THE  HISTORY  OF  INDEX  NUMBERS1 

A  complete  history  of  index  numbers  remains  to  be  written.  Data 
for  it  are  contained  in  C.  M.  Walsh's  Measurement  of  General  Exchange 
Value,  and  are  summarized  in  J.  L.  Laughlin's  Principles  of  Money  and  in 
Wesley  C.  Mitchell's  Index  Numbers  of  Wholesale  Prices,  Bulletin  173  of  the 
United  States  Bureau  of  Labor  Statistics  and  its  revision,  Bulletin  284- 
Here  I  shall  be  even  more  brief,  setting  forth  merely  the  chief  landmarks 
in  the  history  of  index  numbers. 

In  1738  Dutot  published  the  prices  in  the  times  of  Louis  XII  and  of 
Louis  XIV  by  the  formula  here  numbered  51.  That  is,  he  merely  com- 
pared the  sums  of  prices  as  quoted.  In  1747,  as  pointed  out  by  Professor 
Willard  Fisher,2  the  Colony  of  Massachusetts  created  a  tabular  standard 
for  the  payment  of  indebtedness  as  a  means  of  escaping  the  effects  of 
the  depreciation  of  paper  money.  The  same  device  was  re-enacted  in 
1780,  the  state  issuing  notes  "Both  Principal  and  Interest  to  be  paid  in 
the  then  current  Money  of  said  State,  in  a  greater  or  less  Sum,  according 
as  Five  Bushels  of  CORN,  Sixty-eight  Pounds  and  four-seventh  Parts  of  a 
Pound  of  BEEF,  Ten  Pounds  of  SHEEP'S  WOOL,  and  Sixteen  Pounds  of 
SOLE  LEATHER  shall  then  cost,  more  or  less  than  One  Hundred  and 
Thirty  Pounds  current  Money,  at  the  then  current  Prices  of  said  Arti- 
cles." This  is  equivalent  to  Formula  9051,  the  aggregative,  with  arbi- 
trarily chosen  and  constant  weights. 

In  1764  Carli  in  Italy  used  Formula  1,  the  simple  arithmetic  average,  for 
comparing  the  price  levels  of  1500  and  1750  as  revealed  by  the  prices  of 
grain,  wine,  and  oil,  to  show  the  effect  of  the  discovery  of  America  on  the 
purchasing  power  of  money.  In  1798  the  same  formula  was  used,  doubt- 
less independently,  by  G.  Shuckburgh  Evelyn  in  England.  In  1812  Ar- 
thur Young  introduced  weighting  into  Shuckburgh's  method,  thus  using 
Formula  9001.  He  counted  wheat  five  times,  barley  and  oats  twice,  pro- 
visions four  times,  day  labor  five  times,  and  wool,  coal,  and  iron,  once  each. 

The  price  changes  caused  by  the  Napoleonic  wars  and  the  effects  of 
paper  money  led  a  few  students  to  further  studies  in  index  numbers.  In 
1822  Lowe,  and,  in  1833,  Scrope,  both  in  England,  proposed  Formula  9051 ; 
Scrope  says  the  quantities  should  be  "determined  by  the  proportionate 
consumption"  of  the  various  articles.  Lowe  proposed  a  "standard  from 
materials"  reduced  into  tabular  form  which  Scrope  called  "the  tabular 
standard."  This  meant  the  correction  by  means  of  an  index  number  of 
contracts  to  pay  sums  of  money  in  the  future.  In  1853  J.  Prince-Smith 
introduced  the  use  of  algebraic  formulae  into  this  subject,  although  he  did 
not  put  much  trust  in  index  numbers. 

1  These  "  landmarks  "  are,  of  course,  in  addition  to  the  detailed  historical  notes  scattered 
through  the  book,  usually  as  the  concluding  sections  of  the  various  chapters. 
*  "The  Tabular  Standard  in  Massachusetts,"  Quarterly  Journal  of  Economics,  May,  1913. 

458 


APPENDIX  IV  459 

In  1863  Jevons  in  England  used  Formula  21,  the  simple  geometric,  and 
in  1865,  worked  out  index  numbers  for  English  prices  back  to  1782.  He  was 
concerned  chiefly  in  showing  the  "fall  in  the  value  of  gold"  caused  by  the 
outpourings  of  the  gold  mines  beginning  in  1849.  He  endorsed  and  strongly 
urged  Scrope's  proposal  for  a  tabular  standard  of  value.  Jevons  seems  to 
have  been  the  first  to  have  kindled  in  others  an  interest  in  the  subject  and 
may  perhaps  be  considered  the  father  of  index  numbers.  In  1864  Las- 
peyres,  who  in  Germany  worked  out  index  numbers  for  Hamburg  by  For- 
mula 1,  opposed  Jevons'  21  and  proposed  53. 

In  1869  the  London  Economist  began  its  publication  of  index  numbers 
for  22  commodities.  This  still  continues  and  is  the  oldest  of  the  current 
series.  It  uses  Formula  1,  although  the  base  number  is  2200,  instead  of 
100.  Recently  the  number  of  commodities  has  been  doubled. 

In  1874  Paasche  in  Germany  proposed  Formula  54  and  applied  it  to 
22  commodities  for  the  years  1868  to  1872. 

The  fall  of  world  prices  beginning  in  1873,  reversing  the  rise  which  so 
interested  Jevons,  gave  a  new  turn  to  the  study  of  index  numbers.  In 
1880  an  Italian  economist  and  statistician,  Messedaglia,  made  a  commence- 
ment of  studying  the  nature  of  averages  in  application  to  this  subject,  in 
his  II  calcolo  dei  valori  medii  e  le  sue  applicazioni  statistiche.  In  1881, 
H.  C.  Burchard,  Director  of  the  United  States  Mint,  constructed  an  index 
number  for  the  years  1824-1880.  This  seems  to  be  the  first  index  num- 
ber for  the  United  States. 

In  1886  Sauerbeck  presented  a  paper  to  the  Royal  Statistical  Society, 
and  began  his  well-known  series  of  index  numbers  still  continued  by  the 
Statist.  He  used  Formula  1.  In  1886  Soetbeer  began  his  German  series. 
In  1887  and  1889  Edgeworth  wrote  the  two  "Memoranda"  on  index  num- 
bers for  the  British  Association  for  the  Advancement  of  Science,  the  most 
thorough  investigation  of  index  numbers  up  to  that  time.  He  recom- 
mended several  forms  of  index  numbers :  the  arithmetic  average,  both 
weighted  and  simple,  the  simple  median,  and  the  simple  geometric,  ac- 
cording to  the  object  sought.  In  1890  Westergaard  argued  for  the  geo- 
metric mean,  with  simple  or  -constant  weighting  (i.e.  Formula  21  or  9021) 
on  the  ground  of  fulfilling  the  Westergaard,  or  circular  test.  In  1893 
Falkner  in  the  Aldrich  Report  of  the  United  States  Senate  published  index 
numbers  from  1840  to  1891,  using  Formulae  1  and  9001.  In  1897  Brad- 
street's  began  publishing  its  index  number,  using  Formula  51,  the  units 
for  the  various  commodities  being  all  taken  as  one  pound  each. 

The  rise  of  prices  beginning  in  1896,  and  continuing  beyond  the  World 
War,  gave  still  another  stimulus  to  the  study  of  index  numbers.  Beginning 
about  1900,  the  whole  world  increasingly  complained  of  the  high  cost  of 
living,  and  index  numbers  were  increasingly  used  to  measure  the  rising 
tide  of  prices.  In  1901  Walsh  published  his  Measurement  of  General  Ex- 
change Value,  the  largest  and  best  work,  and  the  only  general  treatise  on 
the  theory  of  the  subject  up  to  the  present  time.  In  1901  Dun's  index 
number  by  Formula  53  began.  In  1902  the  United  States  Bureau  of 
Labor  Statistics  began  its  index  number  of  wholesale  prices. 

The  first  index  numbers  were  of  wholesale  prices  and  most  index  numbers 
are  such  today.  For  a  long  time  it  was  thought  that  goods  at  retail,  were 


460         THE  MAKING  OF  INDEX  NUMBERS 

not  sufficiently  standardized  as  to  quality  to  make  retail  index  numbers 
practicable.  This  difficulty  has  not  been  fully  overcome.  But  index 
numbers  of  retail  prices  of  foods  were  begun  in  the  United  States  in  1907, 
and  today  index  numbers  of  retail  prices  are  very  common  in  most  coun- 
tries. Index  numbers  of  wages  are  not  yet  as  fully  developed  as  of  retail 
prices. 

In  1911,  in  my  Purchasing  Power  of  Money,  I  included  a  chapter  and  a 
long  Appendix  on  index  numbers.  In  1912  Knibbs,  the  Statistician  of 
Australia,  urged  Formula  53  on  various  grounds,  especially  ease  of  com- 
putation, and  discussed  the  subject  mathematically.  In  1915  Mitchell 
published  his  thoroughgoing  monograph  on  index  numbers  of  wholesale 
prices,  already  mentioned,  Bulletin  173  of  the  United  States  Bureau  of 
Labor  Statistics  (revised  as  Bulletin  284,  1921). 

In  1918  the  National  Industrial  Conference  Board  published  an  index 
number  of  the  cost  of  living.  In  1919  the  United  States  Bureau  of  Labor 
Statistics  published  an  index  number  of  the  cost  of  living,  including  not 
only  foods,  which  had  hitherto  been  almost  the  only  retail  items  used  in 
index  numbers,  but  substantially  everything  else. 

Thus,  since  the  beginning  of  the  present  century,  index  numbers  have 
spread  very  fast.  In  the  United  States  we  now  have  among  others  the 
index  numbers  of  the  United  States  Bureau  of  Labor  Statistics,  of  the 
Federal  Reserve  Board,  of  Dun's,  of  Bradstreet's,  of  Gibson,  of  the  Times 
Annalist,  of  Babson,  of  the  National  Industrial  Conference  Board,  of  the 
Harvard  Committee  on  Economic  Research,  and  of  the  Massachusetts 
Special  Commission  on  the  Necessaries  of  Life.  A  list,  as  nearly  complete 
as  possible,  of  the  index  numbers,  both  discontinued  and  current,  of  all 
countries  has  already  been  given  in  Appendix  I  (Note  to  Chapter  XVII, 
§14.) 

It  will  be  noticed  that  index  numbers  are  a  very  recent  contrivance. 
That  is,  although  we  may  push  back  the  date  of  their  invention  a  century 
and  three  quarters,  their  current  use  did  not  begin  till  1869  at  the 
earliest,  and  not  in  a  general  way  till  after  1900.  In  fact,  it  may  be 
said  that  their  use  is  only  seriously  beginning  today. 

As  stated  in  the  text,  in  England  the  wages  of  over  three  million  laborers 
have  been  periodically  adjusted  by  means  of  an  index  number. 


APPENDIX  V 

LIST  OF  FORMULAE  FOR  INDEX  NUMBERS 
(For  Reference) 

§  1.  Key  to  the  Principal  Algebraic  Notations 

Po  and  <?o  represent  price  and  quantity  of  a  commodity  at  time  "0"  and 

pi  and  qi  at  time  "1" 
p'o  and  q'o  represent  price  and  quantity  of  another  commodity  at  time  "0" 

and  p'i  and  q\  at  time  "1" 
p"o  and  g"0  represent  price  and  quantity  of  another  commodity  at  time  "0  " 

and  p"i  and  q'\  at  time  "1" 
p'"o  and  q'"o  represent  price  and  quantity  of  another  commodity  at  time  "0  " 

and  p'"i  and  q"\  at  time  "1" 
etc.,  etc. 

Pi  p_i  PI  Qfa  are  prjce  relatives  the  average  of  which  is  POI 
Po  p'o  p"o 


—>  —>  ^—^)  etc.  are  quantity  relatives  the  average  of  which  is  Qoi 
qo  q'o  q  o 

V  is  abbreviation  for 


§  2.  Key  to  Numbering  of  Formulae  of  Index  Numbers 
PRIMARY  FORMULAE   (1-99) 


FORMULA 
No. 

FORMULA 
No. 

1 

31 
5« 
7 
9 

Simple  Arithmetic 

2 

Factor  Antithesis  of    1 

Weighted  I  Arithmetic 
Weighted  II  Arithmetic 
Weighted  III  Arithmetic 
Weighted  IV  Arithmetic 

42 
61 
8 
10 

Factor  Antithesis  of    3 
Factor  Antithesis  of    5 
Factor  Antithesis  of    7 
Factor  Antithesis  of    9 

11 

13 
15 
IT* 

19  2 

Simple  Harmonic 

12 

Factor  Antithesis  of  11 

Weighted  I  Harmonic 
Weighted  II  Harmonic 
Weighted  III  Harmonic 
Weighted  IV  Harmonic 

14 
16 
18  2 

20  » 

Factor  Antithesis  of  13 
Factor  Antithesis  of  15 
Factor  Antithesis  of  17 
Factor  Antithesis  of  19 

1  Reduces  to  Formula  53. 


2  Reduces  to  Formula  54 


461 


462 


THE  MAKING  OF  INDEX  NUMBERS 


FORMULA 
No. 

FORMULA 
No. 

21  1 

Simple  Geometric 

22  2 

Factor  Antithesis  of  21 

23 
25 
27 
29 

Weighted  I  Geometric 
Weighted  II  Geometric 
Weighted  III  Geometric 
Weighted  IV  Geometric 

24 
26 
28 
30 

Factor  Antithesis  of  23 
Factor  Antithesis  of  25 
Factor  Antithesis  of  27 
Factor  Antithesis  of  29 

31  » 

33 
35 
37 
39 

Simple  Median 

32< 

Factor  Antithesis  of  31 

Weighted  I  Median 
Weighted  II  Median 
Weighted  III  Median 
Weighted  IV  Median 

34 
36 
38 
40 

Factor  Antithesis  of  33 
Factor  Antithesis  of  35 
Factor  Antithesis  of  37 
Factor  Antithesis  of  39 

41  • 

Simple  Mode 

42  • 

Factor  Antithesis  of  41 

43 
45 
47 
49 

Weighted  I  Mode 
Weighted  II  Mode 
Weighted  III  Mode 
Weighted  IV  Mode 

44 
46 
48 
50 

Factor  Antithesis  of  43 
Factor  Antithesis  of  45 
Factor  Antithesis  of  47 
Factor  Antithesis  of  49 

51  » 

Simple  Aggregative 

52« 

Factor  Antithesis  of  51 

53 
59" 

Weighted  I  Aggregative 
Weighted  IV  Aggregative 

54 
60  10 

Factor  Antithesis  of  53 
Factor  Antithesis  of  59 

1  Same  as  Formula  121. 
a  Same  as  Formula  122. 
8  Same  as  Formula  131. 
4  Same  as  Formula  132. 


5  Same  as  Formula  141. 

6  Same  as  Formula  142. 

7  Same  as  Formula  151. 

8  Same  as  Formula  152. 


9  Same  as  Formula  54. 
10  Same  as  Formula  53. 


CROSS  FORMULA  FULFILLING  TEST   1    (100-199) 
(All  Crossings  of  Formulae  are  by  Geometric  Mean) 


101   Cross  between    1  and  11 

102  Factor  Antithesis  of 

101  and  cross  between    2  and  12 

1031  Cross  between     3*      ^13 
105  1  Cross  between     5^\/£rl& 
107  Cross  between     £7N>17 
109  Cross  between     U         *19 

104  !  Factor  Antithesis  of 
1061  Factor  Antithesis  of 
108  Factor  Antithesis  of 
110  Factor  Antithesis  of 

103  and  cross  between    4. 
105  and  cross  between    fr^> 
107  and  cross  between    8y* 
109  and  cross  between  10 

& 

\^1B 

121   Cross  between  21  and  21 

122  Factor  Antithesis  of 

121  and  cross  between  22  and  22 

123  Cross  between  23     1 
125  Cross  between  25  \  I 
27  /  [ 
29     J 

124  Factor  Antithesis  of 
126  Factor  Antithesis  of 

123  and  cross  between  24 
125  and  cross  between  26  \ 
28  / 

30   ; 

1 

131  Cross  between  31  and  31 

132  Factor  Antithesis  of 

131  and  cross  between  32  and  32 

133  Cross  between  33 
135  Cross  between  35  \ 
37  J 
39 

134  Factor  Antithesis  of 
136  Factor  Antithesis  of 

133  and  cross  between  34 
135  and  cross  between  36  \ 
38  I 
40 

1  Reduces  to  Formula  353. 


APPENDIX  V 


463 


141   Cross  between  41  and  41 

142  Factor  Antithesis  of  141  and  cross  between  42  and  42 

143  Cross  between  43     1 
145  Cross  between  45  1  1 
47  /  f 
49     J 

144  Factor  Antithesis  of  143  and  cross  between  44     ~| 
146  Factor  Antithesis  of  145  and  cross  between  46  \  1 
48  /  f 
50    J 

151  Cross  between  51  and  51 

152  Factor  Antithesis  of  151  and  cross  between  52  and  52 

1531  Cross  between  53  1 
59  / 

154i  Factor  Antithesis  of  153  and  cross  between  54  1 
60  J 

Reduces  to  Formula  353. 


CROSS   FORMULAE  FULFILLING  TEST  2    (200-299) 


201 

Cross  between    1  and    2 

231  3 

Cross  between  31  and  32 

203  1 

Cross  between    3  and    4 

233 

Cross  between  33  and  34 

205  1 

Cross  between    5  and    6 

235 

Cross  between  35  and  36 

207 

Cross  between    7  and    8 

237 

Cross  between  37  and  38 

209 

Cross  between    9  and  10 

239 

Cross  between  39  and  40 

211 

Cross  between  11  and  12 

241* 

Cross  between  41  and  42 

213 

Cross  between  13  and  14 

243 

Cross  between  43  and  44 

215 

Cross  between  15  and  16 

245 

Cross  between  45  and  46 

217  ' 

Cross  between  17  and  18 

247 

Cross  between  47  and  48 

219  » 

Cross  between  19  and  20 

249 

Cross  between  49  and  50 

221  2 

Cross  between  21  and  22 

251  6 

Cross  between  51  and  52 

223 

Cross  between  23  and  24 

253  • 

Cross  between  53  and  54 

225 

Cross  between  25  and  26 

259  • 

Cross  between  59  and  60 

227 

Cross  between  27  and  28 

229 

Cross  between  29  and  30 

1  Reduces  to  Formula  353. 

2  Same  as  Formula  321. 


8  Same  as  Formula  331. 
*  Same  as  Formula  341. 


8  Same  as  Formula  351. 
6  Reduces  to  Formula  353. 


CROSS   FORMULAE  FULFILLING  BOTH  TESTS    (300-399) 


301 
3031 

Cross  between    1,11;     2,12 

also  between  101  and  102 

also  between  201  and  211 

Cross  between    3,  19  ;     4,  20 

also  between  103  and  104 

also  between  203  and  219 

3051 

Cross  between    5,  17  ;     6,  18 

also  between  105  and  106 

also  between  205  and  217 

307 

Cross  between    7,  15  ;     8,  16 

also  between  107  and  108 

also  between  207  and  215 

309 
321 

Cross  between    9,  13  ;   10,  14 

also  between  109  and  110 

also  between  209  and  213 

Cross  between  21,  21;  22,22 

also  between  121  and  122 

also  between  221  and  221 

323 

Cross  between  23,  29  ;  24,  30 

also  between  123  and  124 

also  between  223  and  229 

Reduces  to  Formula  353. 


464        THE  MAKING  OF  INDEX  NUMBERS 


325 

Cross  between  25,  27  ;  26,  28 

also  between  125  and  126 

also  between  225  and  227 

331 

Cross  between  31,  31  ;  32,  32 

also  between  131  and  132 

also  between  231  and  231 

333 

Cross  between  33,  39  ;  34,  40 

also  between  133  and  134 

also  between  233  and  239 

335 
341 

Cross  between  35,  37  ;  36,  38 

also  between  135  and  136 

also  between  235  and  237 

Cross  between  41,  41  ;  42,  42 

also  between  141  and  142 

also  between  241  and  241 

343 
345 

Cross  between  43,  49  ;  44,  50 

also  between  143  and  144 

also  between  243  and  249 

Cross  between  45,  47  ;  46,  48 

also  between  145  and  146 

also  between  245  and  247 

351 

Cross  between  51,  51  ;  52,  52 

also  between  151  and  152 

also  between  251  and  251 

353 

Cross  between  53,  59  ;  54,  60 

also  between  153  and  154 

also  between  253  and  259 

The  foregoing  formulae  constitute  the  "main  series";  the  following,  the 
"  supplementary  series." 


CROSS  WEIGHT  FORMULA   (1000-1999) 
(Cross  by  Geometric  Mean) 

(1003  and  1013  do  not  fulfill  Test  1 ;  all  1100-1199  fulfill  Test  1  and  1300- 
1399  fulfill  both  tests) 


1003 

Cross  weight  from  3  and   9  ;  also  from   5  and   7 

1004 

Factor  Antithesis  of  1003 

1013 

Cross  weight  from  13  and  19  ;  also  from  15  and  17 

1014 

Factor  Antithesis  of  1013 

1103 

Cross  between  1003  and  1013 

1104 

Factor  Antithesis  of  1103 

1123 

Cross  weight  from  23  and  29  ;  also  from  25  and  27 

1124 

Factor  Antithesis  of  1123 

1133 

Cross  weight  from  33  and  39  ;  also  from  35  and  37 

1134 

Factor  Antithesis  of  1133 

1143 

Cross  weight  from  43  and  49  ;  also  from  45  and  47 

1144 

Factor  Antithesis  of  1143 

1153 

Cross  weight  from  53  and  59 

1154 

Factor  Antithesis  of  1153 

1303 
1323 
1333 
1343 
1353 

Cross  between  1103  and  1104 
Cross  between  1123  and  1124 
Cross  between  1133  and  1134 
Cross  between  1143  and  1144 
Cross  between  1153  and  1154 

APPENDIX  V 


465 


CROSS  WEIGHT   FORMULAE   (2000-4999) 
(Other  than  by  Geometric  Cross) 


2153 
2353 

Cross  weight  (arithmetically)  from  53  and  54 
Cross  between  2153  and  2154 

2154 

Factor  Antithesis  of  2153 

3153 
3353 

Cross  weight  (harmonically)  from  53  and  54 
Cross  between  3153  and  3154 

3154 

Factor  Antithesis  of  3153 

4153 
4353 

Cross  weight  (Lehr's)  from  53  and  54 
Cross  between  4153  and  4154 

4154 

Factor  Antithesis  of  4153 

MISCELLANEOUS  FORMULAE   (5000-9999) 


Crosses  of  Cross  Formulae  (5000-5999) 


5307 

Cross  between  307  and  309 

5323 

Cross  between  323  and  325 

5333 

Cross  between  333  and  335 

5343 

Cross  between  343  and  345 

Broadened  Base  Formulas  (6000-6999) 


6023 
6053 


Like  23  except  that  base  is  average  over  two  or  more  years 
Like  53  except  that  base  is  average  over  two  or  more  years 


Blend  (7000-7999) 


7053     Average  of  353's  reckoned  for  every  year 


Arithmetic  and  Harmonic  Averages  of  Formulae  (8000-8999) 


8053 
8054 


8353 


Simple  arithmetic  average  of  53  and  54 
Simple  harmonic  average  of  53  and  54 
(also  factor  antithesis  of  8053) 
Cross  of  8053  and  8054 


Round  Weight  Formulae  (9000-9999)5 


9051     Calculated  like  51  after  judicious  shifts  of  decimal  points  of  the  36  quotations. 

1  Reduces  to  Formula  353. 

«  For  9001,  9011,  9021,  9031,  and  9041,  none  of  which  are  calculated  in  this  book,  see  §  3 
of  this  Appendix,  Table  62. 


466         THE  MAKING  OF  INDEX  NUMBERS 


§  3.  TABLE  62.   FORMULAE  FOR  INDEX  NUMBERS 
(V  is  abbreviation  for 
ARITHMETIC  TYPES 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

X 

APPROVED  BY 

No. 

Letter 

Name 

1 

A 

Simple 

/SP1> 

Po\ 
I    n    j 

Carli 
Schuckburg- 
Evelyn 
Economist 
Sauerbeck, 
Statist 
Most  others 

2 

*£ 
r+ja. 

n 

3* 

A/ 

Weighted  / 

2Po?o  — 
Spotfo 

U.  S.  Bur.  Labor 
Statistics 

4f 

2907)0  ^ 
Sg-opo 

5t 

A  II 

Weighted  // 

2pogi^ 

2p0?i 

6* 

S^opi  — 
2g0pi 

7 

Kill 

Weighted  /// 

Spi9o| 

2pi?o 

8 
~9~~ 

»M»| 

•jr  4.             Qo 

Stfipo 

A  IV 

Weighted  IV 

Spi^i  ^ 
Spi^i 

Palgrave 

10 

Sgipi  r- 
7-  22. 

2?ipi 

*  Reduces  to  53. 


t  Reduces  to  54. 


APPENDIX  V 


467 


TABLE  62  (Continued) 
HARMONIC  TYPES 


SYMBOLS  FOB  IDENTIFICATION 

FORMULA. 

APPROVED  BY 

No. 

Letter 

Name 

11 

H 

Simple 

^**~            n 

1 

Coggeshall 

12 

"t 

13 

H/ 

Weighted  / 

2p<tfo 

z»| 

14 

r   :     S(?0??0 
S<M>o  ^ 

15 

H// 

Weighted  II 

Spogi 

7>o 
2pogi- 

16 

y  ...    Sg°^ 

^0 

2g0pi  - 

17* 

H//7 

Weighted/// 

Spi^o 

Po 

Sp^o- 

18t 

y    .      S^P° 

'  x^f 

19f 

H/F 

Weighted  IV 

2pi^i 

ZPI^| 

20* 

r    .     2»P» 

90 

2^a- 

*  Reduces  to  53.  t  Reduces  to  54. 


468         THE  MAKING  OF  INDEX  NUMBERS 

TABLE  62  (Continued) 
GEOMETRIC  TYPES 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Letter 

Name 

21* 

22f 

G 

Simple 

Jevons 
Westergaard 
Flux 

-?/£•.£... 

*  PO    p  o 

Nicholson 
Walsh 

7       \q0'7o'" 

23 

G7 

Weighted  7 

^(Sngr-- 

24 

*+*$($-  (gr'~ 

25 
26 

G77 

Weighted  77 

^•PoQij/pi\P°^1/pfi\P'0^'1 

2gopi//<71\9oPi/<7/1\Q'op'i 

27 

G777 

Weighted  777 

2pigo//X>i\  Piflo/p'i\P'i9'o 

28 

V  ^  291W  £iYlpY  iiy  1P  °... 

29 
30 

G7F 

Weighted  IV 

Federal  Reserve 
Board 

^(ffW- 

\\qQ/       \q'0/ 

*  Same  as  121.                               t  Same  as  122. 

APPENDIX  V 


469 


TABLE  62  (Continued) 
MEDIAN  TYPES 


SYMBOLS  FOB  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Letter 

Name 

31* 

Me 

Simple 

Middle  term  of  price 
relatives 

Edgeworth 
Mitchell 

32f 

V  4-  (Middle  term  of 
quantity  relatives) 

33 

Me/ 

Weighted  / 

Mid-weight  term  of 
price  relatives 

34 

V  -5-  (Mid-weight  term  of 
quantity  relatives) 

35 

Me  II 

Weighted  // 

Mid-weight  term  of 
price  relatives 

36 

V  -s-  (Mid-weight  term  of 
quantity  relatives) 

37 

Me/// 

Weighted  /// 

Mid-weight  term  of 
price  relatives 

38 

V  T-  (Mid-weight  term  of 
quantity  relatives) 

39 

Me/F 

Weighted  IV 

Mid-weight  term  of 
price  relatives 

40 

V  -3-  (Mid-weight  term  of 
quantity  relatives) 

*  Same  as  131. 


t  Same  as  132. 


470         THE  MAKING  OF  INDEX  NUMBERS 


TABLE  62  (Continued) 
MODE  TYPES 


STMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Letter 

Name 

41* 

Mo 

Simple 

Commonest  price 
relative 

42f 

V  -;-  (Commonest  quantity 
relative) 

43 

Mo  7 

Weighted  7 

Weightiest  price 
relative 

44 

V  -T-  (Weightiest  quantity 
relative) 

45 

Mo  II 

Weighted  77 

Weightiest  price 
relative 

46 

V  -r  (Weightiest  quantity 
relative) 

47 

Mo  III 

Weighted  777 

Weightiest  price 
relative 

48 

V  -T-  (Weightiest  quantity 
relative) 

49 

Mo  IV 

Weighted  IV 

Weightiest  price 
relative 

50 

V  -T-  (Weightiest  quantity 
relative) 

*  Same  as  141. 


t  Same  as  142. 


APPENDIX  V 


471 


TABLE  62  (Continued) 
AGGREGATIVE  TYPES 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Letter 

Name 

51* 

Ag 

Simple 

ZPI 

Spo 

Bradstreet 
Dutot 

52f 

F-*SL 

Zgo 

Drobisch 
Rawson- 
Rawson 

53 

Ag/ 

Weighted  7 

Spigo 
Spitfo 

Dun 
Fisher 
Knibbs 
Laspejres 
Scrope 
U.  S.  Bur.  Lab. 
Stat. 

54 

v  -j-  2gip° 

2g0po 

Fisher 
Paasche 
Scrope 

59{ 

Ag/F 

Weighted  IV 

Zpigi 
Zpotfi 

60§ 

y^.22ia 

2?OP! 

*  Same  as  151. 
t  Same  as  152. 


J  Same  as  54. 
§  Same  as  53. 


472         THE  MAKING  OF  INDEX  NUMBERS 


TABLE  62  (Continued) 

ARITHMETIC  AND  HARMONIC  CROSSES 
(fulfilling  Test  1) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BT 

No. 

Name 

Cross  of  : 

101 
102 

Simples 

Vl  x  11 

V2  X  12f 

103* 

Weighted 
A  /  &  H  IV 

V3  X  19 

104* 

V4  X  20  f 

105* 

Weighted 
A  II  &  H  /// 

V5  X  17 

106* 

Ve  x  ist 

107 

Weighted 
A  ///  &  H  II 

V7  X  15 

108 

V8  X  16f 

109 

Weighted 
A  IV  &  H  / 

V9X  13 

110 

VlO  X  14f 

*  Reduces  to  353. 

t  Also  the  factor  antithesis  of  the  immediately  preceding  formula,  i.e.  V  -5-  said  pre- 
ceding formula  with  p'a  and  q's  interchanged. 


APPENDIX  V 


473 


TABLE  62  (Continued) 

GEOMETRIC  CROSSES 
(fulfilling  Test  1) 


SYMBOLS  FOB  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Cross  of  : 

V21  X  21 

121* 

Simples 

122  f 

V22  X  22J 

123 

Weighted 
G  /  &  G  IV 

V23  X  29 

124 

V24  X  30* 

125 

Weighted 
G  //  &  G  /// 

V25  X  27 

126 

V26  X  28{ 

*  Reduces  to  21.  t  Reduces  to  22. 

t  Also  the  factor  antithesis  of  the  immediately  preceding  formula,  i.e.  V  •*•  said  pre- 
ceding formula  with  p'a  and  q'a  interchanged. 


MEDIAN  CROSSES 
(fulfilling  Test  1) 


SYMBOLS  FOB  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Cross  of  : 

V31  X  31 

131* 
132  1 

Simples 

V32  X  32t 

133 

Weighted 
Me  I  &  Me  IV 

V33  X  39 

134 

V34  X  40t 

135 

Weighted 
Me  II  &  Me/// 

V35  X  37 

136 

V36  X  38  1 

*  Reduces  to  31.  f  Reduces  to  32. 

t  Also  the  factor  antithesis  of  the  immediately  preceding  formula,  i.e.  V  -s-  said  pre- 
ceding formula  with  p's  and  q'a  interchanged. 


474         THE  MAKING  OF  INDEX  NUMBERS 

TABLE  62  (Continued) 

MODE  CROSSES 
(fulfilling  Test  1) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BT 

No. 

Name 

Cross  of  : 

141* 

Simples 

V41  X  41 

142  f 

V42  X  42  1 

143 

Weighted 
Mo  /  &  Mo  IV 

V43  X  49 

144 

V44  X  50  J 

145 

146 

Weighted 
Mo  II  &  Mo  III 

V45  X  47 

V46  X  48  1 

*  Reduces  to  41.  t  Reduces  to  42. 

t  Also  the  factor  antithesis  of  the  immediately  preceding  formula,  i.e.  V  -5-  said  pre- 
ceding formula  with  p's  and  q's  interchanged. 


AGGREGATIVE  CROSSES 
(fulfilling  Test  1) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Cross  of  : 

V51  X  51 

151* 

Simples 

152  f 

V52  X  52§ 

153  1 

Weighted 
Ag  I  &  Ag  IV 

V53  X  59 

154T 

V54  X  60  § 

*  Reduces  to  51.  t  Reduces  to  52.  I  Reduces  to  353. 

§  Also  the  factor  antithesis  of  the  immediately  preceding  formula,  i.e.  V  +  said  pre- 
ceding formula  with  p's  and  q's  interchanged. 


APPENDIX  V 


475 


TABLE  62  (Continued) 

ARITHMETIC  CROSSES 
(fulfilling  Test  2) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Cross  of  : 

Vl  X2 

201 

Simple 
and  its 
Fact.  Antith. 

203* 

Weighted  A  / 
and  its 
Fact.  Antith. 

V3  X4 

205* 

Weighted  A  // 
and  its 
Fact.  Antith. 

V5~X~6 

207 
209 

Weighted  A  /// 
and  its 
Fact.  Antith. 

V?  X8 

Weighted  A  IV 
and  its 
Fact.  Antith. 

V9  X  10 

*  Reduces  to  353. 


476        THE  MAKING  OF  INDEX  NUMBERS 


TABLE  62  (Continued) 

HARMONIC  CROSSES 
(fulfilling  Test  2) 


SYMBOLS  FOB  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Cross  of  : 

211 

Simple 
and  its 
Fact.  Antith. 

Vll  X  12 

213 

Weighted  H  I  and 
its 
Fact.  Antith. 

Vl3  X  14 

215 

Weighted  H  // 
and  its 
Fact.  Antith. 

Vl5  X  16 

217* 

Weighted  H  /// 
and  its 
Fact.  Antith. 

Vl7  X  18 

219* 

Weighted  H  IV 
and  its 
Fact.  Antith. 

Vl9  X  20 

*  Reduces  to  353. 


APPENDIX  V 


477 


TABLE  62  (Continued) 

GEOMETRIC  CROSSES 
(fulfilling  Test  2) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Cross  of  : 

V21  X  22 

221* 

Simple 
and  its 
Fact.  Antith. 

223 

Weighted  G  7 
and  its 
Fact.  Antith. 

V23  X  24 

225 

Weighted  G  // 
and  its 
Fact.  Antith. 

V25  X  26 

227 

Weighted  GUI' 
and  its 
Fact.  Antith. 

V27  X  28 

229 

Weighted  G  IV 
and  its 
Fact.  Antith. 

V29  X  30 

*  Same  as  321. 


478         THE  MAKING  OF  INDEX  NUMBERS 


TABLE  62  (Continued) 

MEDIAN  CROSSES 
(fulfilling  Test  2) 


SYMBOLS  FOB  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Cross  of  : 

V31  X  32 

231* 

Simple 
and  its 
Fact.  Antith. 

233 

Weighted  Me  / 
and  its 
Fact.  Antith. 

V33  X  34 

235 

Weighted  Me  II 
and  its 
Fact.  Antith. 

V35  X  36 

237 

Weighted  Me  /// 
and  its 
Fact.  Antith. 

V37  X  38 

239 

Weighted  Me  IV 
and  its 
Fact.  Antith. 

V39  X  40 

*  Same  as  331. 


APPENDIX  V 


479 


TABLE  62  (Continued) 

MODE  CROSSES 
(fulfilling  Test  2) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Cross  of  : 

V41  X  42 

241* 

Simple 
and  its 
Fact.  Antith. 

243 

Weighted  Mo  / 
and  its 
Fact.  Antith. 

V43  X  44 

245 

Weighted  Mo  II 
and  its 
Fact.  Antith. 

V45  X  46 

247 

Weighted  Mo  /// 
and  its 
Fact.  Antith. 

"^47  X  48 

249 

Weighted  Mo  IV 
and  its 
Fact.  Antith. 

V49  X  50 

*  Same  as  341. 

AGGREGATIVE  CROSSES 
(fulfilling  Test  2) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Cross  of  : 

V51  X  52 

251* 

Simple 
and  its 
Fact.  Antith. 

253  f 

Weighted  Ag  / 
and  its 
Fact.  Antith. 

V53  X  54 

259  1 

Weighted  Ag  IV 
and  its 
Fact.  Antith. 

V59  X  60 

*  Same  as  351. 


t  Reduces  to  353. 


480         THE  MAKING  OF  INDEX  NUMBERS 


TABLE  62  (Continued) 

ARITHMETIC  AND  HARMONIC  CROSSES 
(fulfilling  both  tests) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Crosses  of  : 

301 

Simples  A  &  H 
and  their 
Fact.  Antith. 

•fyl  X  2  X  11  X  12  or 

VlOl  X  102  or 

V201  X  211 

303* 

Weighted  A  /  &  H  IV 
and  their 
Fact.  Antith. 

\/3  X  4  X  19  X  20  or 

V103  X  104  or 

V203  X  219 

305* 

Weighted  A  II  &  H  /// 
and  their 
Fact.  Antith. 

</5  X  6  X  17  X  18  or 

Vl05  X  106  or 

V205  X  217 

307 

Weighted  A  ///  &  H  II 
and  their 
Fact.  Antith. 

•\/7  X  8  X  15  X  16  or 

V107  X  108  or 

V207  X  215 

309 

Weighted  A  77  &  H  7 
and  their 
Fact.  Antith. 

•\/9  X  10  X  13  X  14  or 

V109  X  110  or 

V209  X  213 

GEOMETRIC  CROSSES 
(fulfilling  both  tests) 


0^51  T 

simple  u 
and  its 
Fact.  Antith. 

-C/21  X  22  X  21  X  22 

or  Vl21  X  122  or 

V221  X  221 

323 

Weighted  G  7  &  G  IV 
and  their 
Fact.  Antith. 

\/23  X  24  X  29  X  30 

or  Vl23  X  124  or 

V223  X  229 

325 

Weighted  G  77  &  G  777 
and  their 
Fact.  Antith. 

^25  X  26  X  27  X  28 

or  V125  X  126  or 

V225  X  227 

*  Reduces  to  353. 


t  Reduces  to  221. 


APPENDIX  V 


481 


TABLE  62  (Continued) 

MEDIAN  CROSSES 
(fulfilling  both  tests) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Crosses  of  : 

331* 

Simple  Me 
and  its 
Fact.  Antith. 

V'Sl  X  32  X  31  X  32 

or  V131  X  132  or 

V231  X  231 

333 

Weighted  Me  /  &  Me  IV 
and  their 
Fact.  Antith. 

\/33  X  34  X  39  X  40 

or  Vl33  X  134  or 

V233  X  239 

335 

Weighted  Me  II  &  Me  III 
and  their 
Fact.  Antith. 

\/35  X  36  X  37  X  38 

or  V135  X  136  or 

V235  X  237 

MODE  CROSSES 
(fulfilling  both  tests) 


341  f 

Simple  Mo 
and  its 
Fact.  Antith. 

•\/41  X  42  X  41  X  42 

or  V141  X  142  or 

V241  X  241 

343 

Weighted  Mo  /  &  Mo  IV 
and  their 
Fact.  Antith. 

-\/43  X  44  X  49  X  50 

or  V143  X  144  or 

V243  X  249 

345 

Weighted  Mo  II  &  Mo  777 
and  their 
Fact.  Antith. 

•\/45  X  46  X  47  X  48 

or  Vl45  X  146  or 

V245  X  247 

*  Reduces  to  231. 


t  Reduces  to  241. 


482 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  62  (Continued) 

AGGREGATIVE  CROSSES 
(fulfilling  both  tests) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

Crosses  of  : 

351* 

Simple  Ag 
and  its 
Fact.  Antith. 

-V/51  X  52  X  51  X  52 

or  Vl51  X  152  or 

V251  X  251 

353  1 

Weighted  Ag/&Ag/F 
and  their 
Fact.  Antith. 

"Ideal" 

Fisher 
Pigou 
Walsh 
Allyn  Young 

VZpitfo  x  ZpiVi 

Spc#o       2p0tfi 

*  Reduces  to  251. 

t  Same  as  103,  104,  105,  106,  153,  154,  203,  205,  217,  219,  253,  259,  303,  305. 

The  foregoing  formulae  constitute  the  "main  series";  the  following,  the 
"supplementary  series." 

CROSS-WEIGHT  ARITHMETICS  AND  HARMONICS 

(fulfilling  neither  test) 


No. 

NAME 

FORMULA 

APPROVED  BT 

Derived  by  : 

svptfo  mi  (^) 

1003 

Crossing 
weights 
of3&9 
or  of  5  &  7 

^Vpoqo  Piqi 

1004 

Fact.  Antith. 
of  1003 

/qi\ 

2Vqopoqipl{-J 

Z^qopo  qipi 

1013 

of  13  &  19 
or  of  15  &  17 

sVpogo  Plql 

SVp0g0  Jhqi  fP°\ 
\pi/ 

1014 

Fact.  Antith. 
of  1013 

2Vg0p0  qipl 

sVg0po  qipi  /<?o\ 
\qi' 

r  APPENDIX  V 

TABLE  62  (Continued) 


483 


No. 

NAME 

FORMULA 

CROSSES  OP  PRECEDING  (fulfilling  Test  1) 

Cross  of  : 

1103 

Cross-weights 
A&H 

V1003  X  1013 

1104 

Fact.  Antith. 
of  1103 

V1004  X  1014 

CROSS-WEIGHT  GEOMETRICS,  MEDIANS,  MODES,  AND  AGGREGATIVES 
(fulfilling  Test  1) 


No. 

NAME 

FORMULA 

APPROVED  BY 

Derived  by  : 

Walsh 

1123 

Crossing 
weights 
of  23  &  29 
or  of  25  &  27 

SVpogo  Pxflil/p!  Wpoflo  pifli 

\w 

1124 

Fact.  Antith. 
of 
1123 

SVflopoQlPll/^Wgopog^ 

\  \qo/ 

1133 

of  33  &  39 
or  of  35  &  37 

Mid  cross-weight  term  of  price 
relatives 

1134 

Fact.  Antith. 
of 
1133 

y  .  Mid  cross-weight  term  of 
quantity  relatives 

1143 

of  43  &  49 
or  of  45  &  47 

Weightiest  cross-weight  price 
relative 

1144 

Fact.  Antith. 
of 
1143 

v  f  Weightiest  cross-weight 
quantity  relative 

1153 

of  53  &  59 

s^Mi  Pi 

Scrope 
Walsh 

2V^po 

1154 

Fact.  Antith. 
of 
1153 

v   m    ^^p^pi  qi 

Walsh 

SVp^ffo 

484 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  62  (Continued) 

CROSSES  OP  PRECEDING  CROSS-WEIGHT  FORMULAE 
(fulfilling  Test  1  and  Test  2) 


SYMBOLS  FOR  IDENTIFICATION 

No. 

Formula 

1303 

V1103  X  1104 

1323 

V1123  X  1124 

1333 

V1133  X  1134 

1343 

V1143  X  1144 

1353 

V1153  X  1154 

CROSS-WEIGHT  AGGREGATIVES,  MISCELLANEOUS 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

(fulfilling  Test  1) 

2153* 

Arithmetically 
crossed  weight 
aggregative 

S*2_b?lpl 

Edgeworth 
Fisher 
Marshall 
Walsh 

^o  +  (7. 

2 

2154* 

Fact.  Antith. 
of 
2153 

2£i±B9l 

y  -i-          z 

Walsh 

i.Po  +  Pin 
2__go 

(fulfilling  Tests  1  and  2) 

2353* 

Cross  of 
preceding 
two 

V2153  X  2154 

*  As  to  alternative  forms,  see  Note  "Alternative  Forms  of  Certain  Formulae"  at  end 
of  table. 


c 


APPENDIX  V 

TABLE  62  (Continued") 


485 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

(fulfilling  Test  1) 

3153* 

Harmonically 
crossed  weight 
aggregative 

/      2      \ 

3(1  +  1)* 

\qo       q\f 

/     2      \ 

z  1+lU 

\?o       W 

3154 

Fact.  Antith. 
of 
3153 

I      2      \ 

2  1+1  ft 

T7      .           \PO            Pi/ 

/_JL_\ 

zfl+ll* 

\PB       Pi/ 

(fulfilling  Tests  1  and  2) 

3353 

Cross  of 
preceding 
two 

V3153  X  3154 

(fulfilling  Test  1) 

4153* 

Weighted 
arithmetically 
crossed  weight 
aggregative 

vP«tfo  +  Ptft  „, 

•"                     fi 

Po  +  Pi 

vpogo  +  Ptfi 

•"                                f  0 

Po  +  Pi 

4154 

Fact.  Antith. 
of 
4153 

Vg0po  +  ftpi 

Lehr 

*-                      yi 

v  .        ^°  +  ft 

v?oPo  +<7ipi 

•"                                yO 

9o  +  qi 

(fulfilling  Tests  1  and  2) 

4353 

Cross  of 
preceding 
two 

V4153  X  4154 

*  As  to  alternative  forms,  see  Note  "  Alternative  Forms  of  Certain  Formulae  "  at  end  of  table. 


486 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  62  (Continued) 

CROSSES  OP  CROSSES 
(fulfilling  Tests  1  and  2) 


No. 

FORMULA 

APPROVED  BY 

5307 

V307  X  309 

5323 

V323  X  325 

5333 

V333  X  335 

5343 

V343  X  345 

GEOMETRIC  AND  AGGREGATIVE  BROADENED  BASE  FORMULAE 
(fulfilling  neither  test) 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

6023 
6023 

Geometric 
Broadened  base 
1913-14 

Same  as  23 
after  substituting 
for  "0,"  0-1,  or'13-'14 

Day 
Persons 

Same 
1913-16 

Same  as  23 
after  substituting 
for  "0,"  0-1-2-3,  or 
'13-'14-'15-'16 

Day 

Persons 

6023 

Same 
1913  and  1918 

Same  as  23 
after  substituting 
for  "0,"  0  and  5,  or 
'13  and  '18 

Day 
Persons 

6053 

Aggregative 
Broadened  base 
1913-14 

Same  as  53 
after  substituting 
for  "0,"  0-1,  or  '13-'14 

6053 
6053 

Same 
1913-16 

Same  as  53 
after  substituting 
for  "0,"  0-1-2-3,  or 
'13-'  14-'  15-'  16 

Same 
1913-18 

Same  as  53 
after  substituting 
for  "0,"  0-1-2-3-4-5,  or 
'13-'14-'15-'16-'17-'18 

APPENDIX  V 


487 


TABLE  62  (Continued) 

ARITHMETIC  AND  HARMONIC  MEANS  OF  AGGREGATIVE  INDEX  NUMBERS 
(fulfilling  neither  test) 


No. 

NAME 

FORMULA 

APPROVED  BY 

7053 

Arithmetic  mean 
of  ideal  formula 
on  different  base 
years 

353  ('13)  +353 
353  ('16)  +353 

('14)  +353  ('15)  + 
('17)  +353  ('18) 

8053 

Arithmetic  mean 
of  aggregative 

2  p\ 
53  +  54  _  Spo 

7o   ,  Sptfc 

Sidgwick 
Drobisch 

2 

2 

8054 

Fact.  Antith. 
of 
8053 

Sgipo  +  2 
T/    .    Sfjopo        2 

9iPi 
tfopi  _            2 

2                  2po9o  i    ^PoQi 
2pi<7o       ^PiQi 

8353* 

V8053  X 

8054 

*  Reduces  to  353. 
ALL  TYPES  OF  INDEX  NUMBERS  WITH  CONSTANT  WEIGHTS 


SYMBOLS  FOR  IDENTIFICATION 

FORMULA 

APPROVED  BY 

No. 

Name 

9001 

Weighted  by 
arbitrary 
constants 

2W  ?1                 where  the  w's 
PQ                 are  arbitrary 
Sw                      constant  weights 

Dun 
Falkner 
Ar.  Young 

Arithmetic 

9011 

Harmonic 

Zw                      where  the  w's 

p0                 are  arbitrary 

2w?  —                  constant  weights 
Pi 

9021 
9031 

Geometric 

^wlipi\w            where  the  w's 
"V  \    /    '  '  *      are  art»itrary 
constant  weights 

Median 

Mid-weight  term  of  price  relatives 

9041 

Mode 

Weightiest  price  relative 

9051 

Aggregative 

Sw  pi                 where  the  w's 

Lowe 

2w  p0                are  arbitrary 
constant  weights 

488         THE  MAKING  OF  INDEX  NUMBERS 


ALTERNATIVE  FORMS  OF  CERTAIN  FORMULAE 

Many  formulae  may  be  changed  into  forms  other  than  those  given  in  the 
foregoing  table.  .  The  footnotes  to  the  table  indicate  some  transformations 
such  as  of  Formula  3  into  Formula  53.  There  are  many  others.  Thus  we 
may  derive  at  least  five  alternative  forms  for  Formula  2153,  five  for  2154, 
two  for  2353,  five  for  3153,  seven  for  4153.  In  most  of  these  cases,  the 
form  easiest  to  calculate  is  not  that  given  in  the  table.  Thus  the  most 
easily  calculated  form  of  2153  is 


S(gi  +  qo)Po 
that  of  2154  is 


and  of  3153 


APPENDIX  VI 

NUMERICAL  DATA  AND  EXAMPLES 

§1.    THE  DATA  FOR  THE  36  COMMODITIES,  PRICES  AND  QUANTITIES 

TABLE  63.    PRICES  OF  THE  36  COMMODITIES,  1913-1918 


No. 

COMMODITY 

Po 
1913 

Pi 
1914 

P« 
1915 

P3 

1916 

pt 
1917 

P6 

1918 

1 

Bacon   

1236 

1295 

1129 

1462 

2382 

2612 

2 

Barley  .          ... 

6263 

6204 

7103 

8750 

1  3232 

1  4611 

3 

Beef 

1295 

1364 

1289 

1382 

1672 

.2213 

4 

Butter 

2969 

2731 

2743 

3179 

4034 

4857 

5 

Cattle 

12  0396 

11  9208 

12  1354 

12  4375 

15  6354 

18  8646 

£>  00  •<!  OS 

Cement      .... 
Coal,  anth.     .     .     . 
Coal,  bit  
Coffee   

1.5800 
5.0636 
1.2700 
1113 

1.5800 
5.0592 
1.1700 
0816 

1.4525 
5.0464 
1.0400 
0745 

1.6888 
5.2906 
2.0700 
0924 

2.0942 
5.6218 
3.5800 
.0929 

2.6465 
6.5089 
2.4000 
.0935 

10 

Coke     

3  0300 

2  3200 

24200 

4  7800 

10.6600 

7.0000 

11 

Copper      .... 

1533 

1318 

1676 

2651 

2764 

.2468 

12 

Cotton 

1279 

1121 

1015 

1447 

2350 

.3178 

13 

Eggs 

2468 

2660 

2597 

2945 

4015 

.4827 

14 

Hav 

11  2500 

12  3182 

11  6250 

10  0625 

17  6042 

21  8958 

15 

Hides 

1727 

1842 

2076 

2391 

2828 

2144 

16 

Hogs     

8.3654 

8.3608 

7.1313 

9.6459 

15.7047 

17.5995 

17 
18 
19 
20 

Iron  bars  .... 
Iron,  pig    .... 
Lead  (white)  .     .     . 
Lead      

1.5100 
14.9025 
.0676 
0437 

1.2000 
13.3900 
.0675 
.0386 

1.3700 
13.5758 
.0698 
0467 

2.5700 
18.6708 
.0927 
.0686 

4.0600 
38.8082 
.1121 
.0879 

3.5000 
36.5340 
.1271 
.0741 

21 
22 
23 
24 

Lumber      .... 
Mutton      .... 
Petroleum      .     .     . 
Pork      

90.3974 
.1025 
.1233 
1486 

90.9904 
.1010 
.1200 
.1543 

90.5000 
.1073 
.1208 
.1429 

91.9000 
.1250 
.1217 
.1618 

105.0400 
.1664 
.1242 
.2435 

121.0455 
.1982 
.1695 
.2495 

25 
26 

Rubber      .... 
Silk             .     .     . 

.8071 
3  9083 

.6158 
4  0573 

.5573 
3  6365 

.6694 
54458 

.6477 
5.9957 

.5490 
6.9770 

27 

Silver               .     . 

5980 

5481 

4969 

6566 

.8142 

.9676 

28 

Skins               .     .     . 

2  5833 

2  6250 

2  7188 

4  1729 

5.5208 

5.5625 

29 
30 
31 
32 

Steel  rails  .... 
Tin,  pig     .     .     «x_^ 
Tin  plate   ...     7 
Wheat 

28.0000 
44.3200 
3.5583 
9131 

28.0000 
35.7000 
3.3688 
1  0412 

28.0000 
38.6600 
3.2417 
1  3443 

31.3333 
43.4800 
5.1250 
1  4165 

38.0000 
61.6500 
9.1250 
2.3211 

54.0000 
87.1042 
7.7300 
2.2352 

33 

Wool 

5883 

5975 

7375 

7900 

1  2841 

1.6600 

34 

1.2500 

1.2500 

1.2396 

1.4050 

1.7604 

2.3000 

35 

Lard 

1101 

1037 

0940 

1347 

.2170 

.2603 

36 

Oats      .     .     .  '  .     . 

.3758 

.4191 

.4958 

.4552 

.6372 

.7747 

490         THE  MAKING  OF  INDEX  NUMBERS 


TABLE  64.    QUANTITIES  MARKETED  OP  THE  36  COMMODITIES,  1913-1918 
(in  millions  of  units) 


No. 

COMMODITY 

qo 
1913 

H 
1914 

«2 

1915 

qa 
1916 

«4 

1917 

qs 
1918 

1 

Bacon,  Ib  

1077. 

1069. 

1869. 

1481. 

1187. 

1498. 

2 

Barley,  bu.     .     .     . 

178.2 

195. 

228.9 

182.3 

209. 

256.4 

3 

Beef,  Ib  

6589. 

6522. 

6820. 

7134. 

8417. 

10244. 

4 

Butter,  Ib.      ... 

1757. 

1780. 

1800. 

1820. 

1842. 

1916. 

5 

Cattle,  owt.    .     .     . 

69.8 

67.6 

71.5 

83.1 

103.5 

118.3 

6 

Cement,  bbl.  .     .     . 

85.8 

84.4 

84.4 

92. 

88.1 

69.4 

7 

Coal,  anth.,  ton  .     . 

6.9 

6.86 

6.78 

6.75 

7.83 

7.69 

8 

Coal,  bit.,  ton     .     . 

477. 

424. 

443. 

502. 

552. 

583. 

9 

Coffee,  Ib  

863. 

1002. 

1119. 

1201. 

1320. 

1144. 

10 

Coke,  short  ton  .     . 

46.3 

34.6 

41.6 

54.5 

56.7 

55. 

11 

Copper,  Ib.     .     .     . 

812.3 

620.5 

1043.5 

1429.8 

1316.5 

1648.3 

12 

Cotton,  Ib.     .     .     . 

2785. 

2820. 

2838. 

3235. 

3423. 

3298. 

13 

Eggs,  doz.       .     .     . 

1722. 

1759. 

1791. 

1828. 

1882. 

1908. 

14 

Hay,  ton    .... 

79.2 

83. 

103. 

111. 

94.9 

89.8 

15 

Hides,  Ib  

672. 

924. 

1227. 

1212. 

1113. 

663. 

16 

Hogs,  cwt.    .      .     . 

68.4 

65.1 

76.8 

8R.2 

67.8 

82.4 

17 

Iron  bar,  cwt.     .     . 

79.2 

50.4 

82.6 

132.4 

133. 

132. 

18 

Iron,  pig,  ton      .     . 

31. 

23.3 

29.9 

39.4 

38.7 

38.1 

19 

Lead  (white),  Ib.     . 

286. 

318. 

312. 

258. 

230. 

216. 

20 

Lead,  Ib  

823.7 

1025.6 

1014.1 

1104.5 

1099.8 

1083. 

21 

Lumber,  M  bd.  ft.  . 

21.8 

20.7 

20.5 

22.3 

21.2 

19.2 

22 

Mutton,  Ib.    .     .    . 

732. 

734. 

629. 

618. 

474. 

513. 

23 

Petroleum,  gal.  .     . 

10400. 

11200. 

11840. 

12640. 

14880. 

15680. 

24 

Pork,  Ib  

9211. 

8871. 

9912. 

10524. 

8427. 

11426. 

25 

Rubber,  Ib.     .     .     . 

115.8 

136.6 

231.4 

258.8 

375.9 

351.5 

26 

Silk,  Ib  

19.1 

19.1 

20. 

24.4 

29.4 

27.1 

27 

Silver,  oz  

146.1 

144. 

173.4 

139.3 

133.6 

140.7 

28 

Skins,  skin      .     .     . 

6.7 

5.9 

4.3 

5.6 

2.7 

.7 

29 

Steel  rails,  ton    .     . 

3.5 

1.95 

2.2 

2.86 

2.94 

2.37 

30 

Tin,  pig,  cwt./    .     . 

1.04 

.95 

1.16 

1.43 

1.56 

1.59 

31 

Tin,  plate,  cwt.  .     . 

15.3 

17.3 

19.7 

22.8 

29.5 

28. 

32 

Wheat,  bu.     .     .     . 

555. 

654. 

588. 

642. 

605. 

562. 

33 

Wool,  Ib.   ...     .     . 

448. 

550. 

699. 

737. 

707. 

752. 

34 

Lime,  bbl.,  300  Ib.  . 

23.3 

22.5 

25. 

27.1 

24. 

20.2 

35 

Lard,  Ib  

1100. 

955. 

1050. 

1141. 

927. 

1107. 

36 

Oats,  bu  

1122. 

1240. 

1360. 

1480. 

1587. 

1538. 

§2.  EXAMPLES,  IN  TABULAR  FORM,  SHOWING  HOW  TO  CALCULATE  INDEX 
NUMBERS  BY  THE  NINE  MOST  PRACTICAL  FORMULAE 


The  following  nine  model  examples  may  be  of  assistance  to  the  reader 
who  desires  practical  and  specific  directions  for  calculating  an  index  num- 
ber. They  include  all  of  the  eight  formulae  mentioned  in  Chapter  XVII, 
§  8,  as  the  formulae  most  recommended  for  practical  use,  together  with 
8053,  a  makeshift  for  353.  Formulae  53,  54,  and  8053,  are  given  first  and 
are  followed  by  the  others  in  the  same  order  as  in  Chapter  XVII,  §  8. 


APPENDIX  VI  491 

In  each  case  the  data  used  are  those  for  the  36  commodities  as  given  on 
the  two  preceding  pages. 

Formula  53,  Laspeyres',  Aggregative  I,    Poi 

(For  discussion  see  pp.  56-60,  131-2,  237-40) 
Computation  of  2po<?o 

PER  UNIT  MILLION  UNITS 

1  (Bacon);    p0  =  $0.1236  ;  q0  =  1077. ;      p<#0  =.  1236  X  1077    =133.117 

2  (Barley) ;  p'Q  =      .6263  ;  q'0  =    178.2;   pV0  =  .6263  X    178.2  =  111.607 

3  (Beef)  P'Vo  =  •  1295  X  6589    =853.276 

4  .2969X1757    =521.653 


36 

.3758  X  1122 

=  421.648 

(adding)         Sp<#0  = 

13104.818 

Computation  of  Spi^o 

1 

Pi  =      .1295;  go  =  1077;        2^0  =  .  1295  X  1077 

=  139.47 

2 

p'ig'o  =  .6204  X    178.2  =  110.56 

3 

.1364X6589 

=  898.74 

36  .4191  X  1122    =470.23 

(adding)         Spig0  =  13095.78 

Whence 

T-n  n  I^OQ^  7Q 

P01  =  .flPill  =_i£M^iL2_  =    99.93  per  cent  =  index  number  for  1914 
Zpotfo       13104.818 

Likewise 

p°-^=Hir 99-67  -  "=  "   "   "I9is 


Likewise 
P03  =  ^P^0  =  14950.13    =  114Q8    „      „    =     „          „ 

Sp0«o       13104.818 
Likewise 

P04  =  sP4go  =  Ji1238-49   -=  162.07    "      "    =     "          "          »   1917 
13104.818 


Likewise 

P05=^Ml=23308-95    =177.87    »      »    =     "         »         »   1918 
13104.818 


The  above  is  by  the  fixed  base  system. 


492         THE  MAKING  OF  INDEX  NUMBERS 

By  the  chain  system,  we  have 

=  13095.78    =    99.93  per  cent 

13104.818 
13059.052  =  1Q()  2Q    „       „ 


13033.034 
16233.560 
14280.976 


113.67 


p     =  25388.869  =  M2  ?2    „       „ 

17789.440 
p  =  27690.677  =  „  „ 

25191.136 
Whence,  by  successive  multiplication 

Poi  =  99.93 

=  99.93  per  cent  =  index  number  for  1914 
PoiPi2  =  99.93  X  100.20 

=  100.13  per  cent  =  index  number  for  1915 
PoiP12P23  =  99.93  X  100.20  X  113.67 

=  113.82  per  cent  =  index  number  for  1916 
PoiPi2P23P34  =  99.93  X  100.20  X  113.67  X  142.72 

=  162.44  per  cent  =  index  number  for  1917 
PoiPi2P23P34P45  =  99.93  X  100.20  X  113.67  X  142.72  X  109.92 
=  178.56  per  cent  =  index  number  for  1918 

Formula  54,  Paasche's,  Aggregative  IV,  Poi  = 


(For  discussion  see  pages  cited  for  Formula  53,  especially  pp.  131-2) 
Computation  of  *2p\q\ 

1  pi  =  -1295  31  =  1069.       ptfi    =  .1295  X  1069.  =  138.436 

2  p'lq'i  =  .6204  X    195.  =  120.978 

3  .1364  X  6522.  =  889.601 


36 

519.684 

(adding) 
Computation  of  Spo^i 

2 

Zpiffi                                 13033.034 

p0qi  =  .1236  X  1069.  =  132.13 
.6263  X    195.  =  122.13 

36 
(adding) 
Whence 
Pni   _2Piqi       13033.034.  ,   inna. 

465.99 
Spo3i                                 12991.81 

npr  r>pnt.   =  inrlp-5r  rmmV»pr  for  1914 

12991.81 


APPENDIX  VI  493 

S-»<>09       14280  976 
P02  =  _J^l  = : =  100.10  per  cent  =  index  number  for 

2p0q2       14266.81 

p     =  17789.440  =  114  35    „       „  „ 

15557.52 

P04  =  161.05    "      "    =      "          "          "   1917 

P05  =  177.43    "        '    =      "  "   1918 

The  chain  figures  in  this  and  subsequent  examples  may  be  derived,  as 
in  the  previous  example,  by  linking.  Thus  PoiPi2P23  =  100.32  X  100.01 
X  114.45  =  114.83  per  cent  =  index  number  for  1916. 


Formula  8053,  Pa  =  (53)  +  <•<*>  = 

(For  discussion  see  pp.  174-7) 
P01  =  99-93  +  100.32  =  100;12  =  index  number  for 

P02  =  99.89  =     "         "         "   1915 

etc. 


Formula  853,  "Ideal,"  P0i  =  V(53)  X  (54)  =        M2  x 

\  Spogo 

(For  discussion  see  pp.  220-9,  234-42) 


Poi  =  V99.93  X  100.32  =  100.12  =  index  number  for  1914 
P02  =  99.89  =     "          "          "   1915 

etc. 

The  square  root  may  be  extracted  "by  hand/'  by  logarithms,  or  (most 
quickly),  by  a  calculating  machine,  in  which  case  the  total  time  required 
to  calculate  the  five  figures  (fixed  base)  is  14.3  hours.  But  it  is  seldom, 
if  ever,  necessary  actually  to  extract  the  square  root  because  the  two  figures 
under  the  radical  are  always  so  close  together  that  the  preceding  Formula 
8053  (requiring  14.1  hours)  can  be  used  instead. 

The  results  of  8053  and  353  agree  to  the  second  decimal  place,  provided 
53  and  54  do  not  differ  by  more  than  1  per  cent,  which  is  usually  the  case. 
Whether  or  not  they  so  differ  can  always  be  seen  at  a  glance.  In  case  they 
do  differ  by  more  than  1  per  cent  and  the  calculator  still  wishes  to  avoid 
the  process  of  root  extraction  he  can  almost  as  quickly  get  the  result  by 
"trial  and  error,"  using  8053  as  a  basis. 

Thus,  let  53  =  101.22  per  cent  and  54  =  104.26  per  cent.  Their  dif- 
ference 3.04  exceeds  1  per  cent  (which  would  be  1.0122).  We  find  8053  = 
01.22  +  104.26  e  geometric 


we  seek,  is  slightly  smaller.     We  therefore  try  102.73  by  comparing  its 
square  ([102.73]2  =  105.535  per  rwnt)  with  what  it  should  be  (i.e.  101.22  X 


494         THE  MAKING  OF  INDEX  NUMBERS 

104.26  =  105.532  per  cent).     Here  the  square  is  slightly  too  great  but  is 
nearer  than  the  square  of  102.72,  which  is  105.514  per  cent.     Therefore 
102.73  is  the  result  sought. 
A  second  and  more  systematic  method  of  avoiding  root  extraction  is 

to  calculate   8053  =  102.74  and   8054  =  -j—  ?  —  —  =  —  -  -  -  — 

(53)  +  (54)       101.22  +  104.26 

=  102.72.  The  geometric  mean  of  these  two  is  necessarily  353*;  but 
these  two  (8053  and  8054)  will  always  be  within  1  per  cent  of  each  other, 
(even  if  the  original  53  and  54  differ  by  as  much  as  25  per  cent),  so 
that  their  arithmetic  mean  (here  102.73  per  cent)  will  always  be  accurate 
to  the  second  decimal  place. 


Formula  2158,  Edgeworth-Marshall's  Aggregative,  Pol  =  2(gf°  + 

2(?o  +  qi)po 

(For  discussion  see  pp.  194-5,  401-7,  428-30) 

This  is*  usually  t  a  sufficiently  accurate  makeshift  for  353  and  requires  9.6 
hours  as  against  14.1  hours  for  8053  and  14.3  hours  for  353. 
Computation  of  S(g0  +  <?i)pi 

1  («o  +  «i)pi  =  (1077.    +  1069.  )  X  .1295  =  277.9070 

2  (  178.2  +    195.0)  X  .6204  =  231.5333 


36  =    989.9142 

(adding)     S(g0  +  9i)pi  =  26128.814 

(similarly)  S(g0  +  3i)po  =  26096.628 

9R19R  Q1J. 

Whence  P0i  =          a         =  100.12  per  cent  =  index  number  for  1914. 
26096.628 

Likewise  P02  =    99.89  per  cent  =  index  number  for  1915. 

etc. 

Formula  6053  (for  discussion  see  pp.  312-3,  318-20)  (assuming  1913- 
1914  the  "broadened  base")  is  derived  exactly  as  2153  above  except  that 
go  +  <Zi  is  retained  throughout  all  five  computations  instead  of  changing  to 
go  +  <?2  in  computing  P02,  etc.  If  1913- '14- '15  is  the  broadened  base, 
So  +  <?i  +  #2  is  so  used. 

Formula  63  has  already  been  exemplified. 

Formula  9051,  ^2i  (for  discussion  see  pp.  198,  327-8,  348)  is  like  53 

Siypo 

except  that  the  IP'S  replace  the  g's  and  are  round  numbers  (1,  10,  100, 
etc.).  These  factors  merely  shift  the  decimal  points  of  the  p's  so  that 
Formula  9051  is  really  Formula  51  with  such  shifts,  each  shift  being  the 
best  round  guess  at  the  proper  factor. 

*See  Appendix  I  (Note  to  Chapter  IX,  §  1). 
t  See  Appendix  I  (Note  to  Chapter  XV,  §  2). 


APPENDIX  VI  495 

1  pi  =  .1295;  w  =  1000;  wpi  =  1000  X  .1295  =    129.5 

2  p\  =  .6204;  100  X  .6204  =      62.04 

3  .1364  1364. 


36  419.1 


(adding)          St^  =  12697.242 

Likewise          Sipp0  =  12487.4043 

Whence  Pn  =  12697'242  =  101.68 

12487.4043 

SimUarly  P^  =  103.10 
etc. 


Formula  21,  Simple  Geometric,  PQl  = 

Pop  op    o 

(For  discussion  see  pp.  33-5,  211-2,  260-4) 

1  log  pi   =  log  .1295  =  1.11227 

2  log  p'i  =  log  .6204  =  1.79267 

3  log  .1364  =  1.13481 

4  1.43632 


36  1.62232 


(adding)  Slogpi  2.13755 

Similarly  S  log  p0  2.81385 

(subtracting)  1.32370  =  35.32370  -  36 

(dividing  by  n  =  36)  .98121  -    1  =  1.98121 

which  is  the  log  of  Poi  =  95.77  per  cent 

Similarly                 P02  =  96.79    "       " 

etc. 

Avoiding  logarithms.  The  many  users  of  index  numbers  who  wish  to 
avoid  logarithms  and  geometric  means,  such  as  Formula  21,  may  use  the 

formula  — -.    This  is  practically  coincident  with  Formula  101  and 

2 

so  with  21. 

A  somewhat  similar  remark  applies  when  the  problem  is  how  best,  with- 
out recourse  to  logarithms,  to  utilize  rough  weights  in  averaging  two  or 
more  price  relatives,  or  two  or  more  index  numbers  already  supplied.  Sup- 
pose, for  instance,  we  wish  to  calculate  an  index  number  for  "the  general 
level  of  prices"  by  combining  existing  index  numbers  of  (1)  wholesale  com- 
modity prices,  (2)  retail  commodity  prices,  (3)  prices  of  shares  on  the  Stock 
Exchange,  and  (4)  wages,  assuming  that  the  separate  index  numbers  of 
(1),  (2),  (3),  (4)  are,  respectively,  200,  150,  250,  125,  and  that  their  rough 


496         THE  MAKING  OF  INDEX  NUMBERS 

weights  (representing,  say,  their  roughly  estimated  values  in  exchange 
during  a  series  of  years)  are  10,  5,  3,  1.  The  arithmetic  formula 

10  X  2.00  +  5  X  1.50  +  3  X  2.50  +  1  X  1.25  = 
10  +  5+3  +  1 

(practically  Formula  1003)  would  be  improper,  having  an  appreciable 
upward  bias  because  the  200,  150,  250,  125  disperse  widely ;  the  harmonic 
formula 

10+5+3  +  1 1.8387 


(practically  Formula  1013)  would  be  improper  for  the  opposite  reason; 
the  geometric  formula 

1-y/(2.00)10  X  (1.50)5  X  (2.50)3  X  (1.25) 

would  be  the  best,  but  requires  logarithms;  the  aggregative  is  im- 
practicable, since  our  weights,  which  are  values,  cannot  be  translated 
into  quantities.  We  have  recourse,  then,  to  an  average  of  the  first  two 
above  —  what  is  practically  Formula  1103,  i.e.  we  take  the  above  arith- 
metic and  harmonic  averages,  namely  1.9079  and  1.8387,  and  average  them 
arithmetically,  obtaining  1.8733.  Or,  instead  of  resting  content  with  this 
result,  we  could  (though  it  would  seldom  if  ever  be  worth  while)  proceed 
another  step  by  also  averaging  the  1.9079  and  1.8387  harmonically  and 
then  taking  the  arithmetic  average  of  the  two  results  (1.8733  and  1.8727), 
which  is  1.8730,  and  so  on,  if  desired,  to  any  number  of  stages,  thereby  ap- 
proximating the  geometric  mean  of  1.9079  and  1.8387  as  closely  as  we  wish. 

Formula  31,  Simple  Median,  mid-term  among  the  price  relatives,  — ,  — ,  ... 

Po    p'o 

(For  discussion  see  pp.  35-6,  209-12,  260-4) 

104.77  per  cent 
po  .1236 

2  -2J  -  '—  =    99.06    "      " 

p'o  .6263 

Rearranging  these  36  price  relatives  in  the  order  of  their  magnitudes,  we 

find 

lowest  price  relative  (coffee)  73.32  per  cent 

next  lowest  price  relative  (rubber)          76.30   "      " 


18th  (barley)  99.06    ' 

19th  (white  lead)     99.85    " 


highest  (wheat)         114.03    "    " 


APPENDIX  VI  497 

The  median  lies  between  the  two  middlemost  terms,  the  18th  and  19th, 
99.06  and  99.85,  and  is  most  simply  taken  as  their  arithmetic  mean  (al- 
though most  properly  their  geometric  mean)  P0i  =  99.45 

Similarly  P02  =  98.57 

etc. 

A  little  time  may  be  saved  by  not  rearranging  the  order  of  terms  but 
crossing  off  from  the  original  list  any  pair  of  terms,  one  very  high  and  one 
very  low  so  as  to  make  sure  that  they  are  on  opposite  sides  of  the  median ; 
then  likewise  erase  another  pair  of  extreme  terms,  i.e.  two  which  surely 
lie  astride  of  the  median,  and  so  on  until  so  few  terms  are  left  that  the  me- 
dian is  obvious. 

Another  practical  index  number,  calculated  partly  by  Formula  53  and 
partly  by  Formula  9051,  is  described  on  p.  346.  Formula  1  (simple  arith- 
metic) is  exemplified  on  pp.  15-24  but  is  not  recommended  for  practical 
use.  Formula  3  (base  weighted  arithmetic)  is  best  reduced  to  Formula  53 
before  calculating. 


APPENDIX  VII 


TABLE  65.  INDEX  NUMBERS  BY  134  FORMULAE  FOR  PRICES 
BY  THE  FIXED  BASE  SYSTEM  AND  (IN  NOTEWORTHY 
CASES)  THE  CHAIN  SYSTEM 

(1913  =  100) 

Although  only  the  specified  Price  indexes  are  here  given,  Quantity  indexes~as  well  as 
Price  indexes  —  both  fixed  base  and  chain  —  have  been  computed  for  all  the  134  formulae 
and  are  utilized  in  the  charts. 

PRIMARY  FORMULAE   (1-99) 

Those  for  which  figures  are  given  conform  to  neither  test. 
ARITHMETIC 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

IANKS*  OF  FIRST 
20  IN  ACCURACY, 
SPEED,     SIMPLIC- 
TY  OF   FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

1 

Fixed 
Chain 

96.32 
96.32 

98.03 
97.94 

123.68 
125.33 

175.79 
175.65 

186.70 
193.42 

3rd  in  speed 
3rd  in  sim- 
plicity 

2 

Fixed 
Chain 

100.18 
100.18 

95.93 
95.47 

109.71 
107.83 

152.75 

152.42 

177.13 

177.69 

15th  in  speed 

(3) 

Same  as  53  (necessarily) 

(4) 

Same  as  54  (necessarily) 

(5) 

Same  as  54  (necessarily) 

(6) 

7 

Same  as  53  (necessarily) 

Fixed 

100.55 

101.77 

117.77 

180.53 

186.98 

8 

Fixed 

99.02 

97.36 

111.45 

152.42 

167.06 

9 

Fixed 
Chain 

100.93 
100.93 

102.33 
102.10 

118.29 
122.  41 

180.72 

130.40 

187.18 
205.56 

10 

Fixed 

98.70 

96.97 

111.10 

154.96 

169.27 

*  As  revised  in  Chapter  XVI,  §  9. 
498 


APPENDIX  VII 


499 


TABLE  65  (Continued) 
HARMONIC 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

11 

Fixed 
Chain 

95.19 
95.19 

95.58 
95.64 

119.12 
117.71 

157.88 
15847 

171.79 
167.76 

3rd  in  speed 
9th  in  sim- 
plicity 

12 

Fixed 
Chain 

103.48 

10348 

101.31 
101.97 

115.35 

117.72 

172.11 

172.55 

243.67 
217.65 

15th  in  speed 

13 

Fixed 
Chain 

99.26 
99.26 

97.84 
9845 

111.01 
108.19 

147.19 

148.14 

168.59 
157.78 

8th  in  speed 

14 

Fixed 

101.81 

102.41 

116.80 

168.37 

189.80 

15 

Fixed 

99.65 

98.11 

111.02 

144.97 

166.85 

16 

Fixed 

101.34 

101.98 

116.63 

168.60 

189.38 

(17) 
(18) 

Same  as  53  (necessarily) 

Same  as  54  (necessarily) 

(19) 

Same  as  54  (necessarily) 

(20) 

Same  as  53  (necessarily) 

*  As  revised  in  Chapter  XVI,  §  9. 


500 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  65  (Continued) 
GEOMETRIC 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OF  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

21 

Same  as  121  (necessarily) 

22 

Same  as  122  (necessarily) 

23 

Fixed 
Chain 

99.61 
99.61 

98.72 
99.28 

111.45 
110.91 

154.08 
155.03 

173.30 
166.93 

17th  in  speed 
18th   in  sim- 
plicity 

24 

Fixed 

101.02 

101.32 

115.64 

164.85 

182.84 

25 

Fixed 

99.99 

99.07 

112.58 

152.45 

172.37 

26 

Fixed 

100.60 

100.88 

115.42 

165.37 

182.61 

27 

Fixed 

100.25 

100.67 

115.82 

170.82 

182.45 

28 

Fixed 

99.65 

98.82 

112.98 

157.09 

172.27 

29 

Fixed 

100.63 

101.17 

116.26 

170.44 

182.41 

30 

Fixed 

99.29 

98.41 

112.67 

158.70 

173.60 

*  As  revised  in  Chapter  XVI,  §  9. 


APPENDIX  VII 


501 


TABLE  65  (Continued) 
MEDIAN 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OF  FIRST 
20  IN  ACCURACY, 
SPEED,     SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

31 

Same  as  131  (necessarily) 

32 

Same  as  132  (necessarily) 

33 

Fixed 
Chain 

100.34 

100.34 

99.39 
99.70 

107.17 
106.80 

156.12 

150.22 

169.14 

173.34 

16th  in  speed 

34 

Fixed 

101.20 

104.66 

117.57 

165.53 

181.97 

35 

Fixed 

100.48 

99.41 

107.37 

160.18 

169.14 

36 

Fixed 

100.97 

104.01 

117.62 

165.49 

182.16 

37 

Fixed 

100.61 

99.65 

108.77 

163.84 

188.25 

38 

Fixed 

100.57 

102.07 

116.74 

157.84 

179.74 

39 

Fixed 

100.75 

99.97 

109.08 

163.84 

178.12 

40 

Fixed 

100.52 

101.78 

116.85 

159.90 

180.33 

*  As  revised  in  Chapter  XVI,  §  9. 


502 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  65  (Continued) 
MODE 


IDENTI- 
FICATION 

NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,    SIMPLIC- 
ITY OP  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

41 

Same  as  141  (necessarily) 

42 

Same  as  142  (necessarily) 

43 

Fixed 

101. 

100. 

108. 

164. 

168. 

44 

Fixed 

103. 

106. 

132. 

196. 

180. 

45 

Same  figures  as  for  43 

46 

Same  figures  as  for  44 

47 

Same  figures  as  for  43 

48 

Same  figures  as  for  44 

49 

Same  figures  as  for  43 

50 

Same  figures  as  for  44 

*  As  revised  in  Chapter  XVI,  §  9. 


APPENDIX  VII 


503 


TABLE  65  (Continued) 
AGGREGATIVE 


ID  ENTI- 
FICATION 

NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO    CIRCULAR 
TEST 

51 

Same  as  151  (necessarily) 

52 

Same  as  152  (necessarily) 

53  f 

Fixed 
Chain 

99.93 
99.93 

99.67 
100.13 

114.08 
113.82 

162.07 
162.44 

177.87 
178.56 

4th  in  speed 
5th  in  sim- 
plicity 

54J   ' 

Fixed 
Chain 

100.32 
100.32 

100.10 
100.33 

114.35 
114-83 

161.05 
162.02 

177.43 

178.43 

13th  in  speed 
6th  in  sim- 
plicity 

59 

Same  as  54  (necessarily) 

60 

Same  as  53  (necessarily) 

*  As  revised  in  Chapter  XVI,  §  9. 
t  53  -  3,  6,  17,  20,  60. 
1 64  -  4,  5,  18,  19,  59. 


504 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  65  (Continued) 

CROSS  FORMULA   (100-199) 

Those  for  which  figures  are  given  fulfill  Test  1  only. 

ARITHMETIC  AND  HARMONIC  CROSSES 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

101 

Fixed 
Chain 

95.75 
95.75 

96.80 
96.78 

121.38 
12146 

166.60 
166.84 

179.09 
180.13 

9th  in  speed 

102 

Fixed 

101.81 

98.58 

112.50 

162.14 

207.75 

103 

Same  as  353  (necessarily) 

104 

Same  as  353  (necessarily) 

105 

Same  as  353  (necessarily) 

106 

Same  as  353  (necessarily) 

107 

Fixed 

100.10 

99.92 

114.35 

161.78 

176.63 

108 

Fixed 

100.17 

99.64 

114.01 

160.31 

177.87 

109 

Fixed 

100.09 

100.06 

114.59 

163.10 

177.64 

110 

Fixed 
Chain 

100.24 
100.24 

99.65 
100.18 

113.91 
114.14 

161.53 
162.06 

179.24 

178.52 

*  As  revised  in  Chapter  XVI,  §  9. 


APPENDIX  VII 


505 


TABLE  65  (Continued) 
GEOMETRIC  CROSSES 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

121 

(21) 

Fixed 
Chain 

95.77 
San 

96.79 
le  as  Fix 

121.37 
ed  Base 

166.65 
'necessar 

180.12 

iiy) 

6th  in  speed 
10th  in  sim- 
plicity 
1st  in  con- 
formity 

122 

(22) 

Fixed 
Chain 

101.71 
San 

98.62 
le  as  Fix 

112.60 
ed  Base 

161.88 
[necessar 

194.14 

iiy) 

18th  in  speed 
1st  in  con- 
formity 

123 

Fixed 
Chain 

100.12 
100.12 

99.94 
100.24 

113.83 
114-68 

162.05 
162.75 

177.80 
178.87 

15th  in  ac- 
curacy 

124 

Fixed 
Chain 

100.16 
100.16 

99.85 
100.23 

114.25 

114.26 

161.74 

162.18 

178.16 
178.60 

17th  in  ac- 
curacy 

125 

Fixed 
Chain 

100.12 
100.12 

99.87 
100.24 

114.19 

114-33 

161.37 
162.18 

177.34 

178.36 

14th  in  ac- 
curacy 

126 

Fixed 

Chain 

100.12 
100.12 

99.85 
100.22 

114.20 
114-66 

161.18 

162.64 

177.36 

178.81 

16th  in  ac- 
curacy 

*  As  revised  in  Chapter  XVI,  §  9. 


506 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  65  (Continued) 
MEDIAN  CROSSES 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY 
SPEED,  SIMPLIC- 
ITY OP  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

131 

(31) 

Fixed 
Chain 

99.45 
99.45 

98.57 
99.33 

118.81 
117.50 

163.81 
165.86 

190.92 
180.07 

10th  in  speed 
4th  in  sim- 
plicity 

132 

(32) 

Fixed 

100.11 

102.20 

116.01 

162.15 

183.54 

133 

Fixed 

100.54 

99.68 

108.12 

159.93 

173.57 

134 

Fixed 

100.86 

103.21 

117.21 

162.69 

181.15 

135 

Fixed 

100.54 

99.53 

108.07 

162.00 

178.44 

136 

Fixed 

100.77 

103.04 

117.18 

161.62 

180.95 

*  As  revised  in  Chapter  XVI,  §  9. 


MODE  CROSSES 


IDENTI- 
FICATION 

NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

141 

(41) 

Fixed 
Chain 

98. 
98. 

98. 
95. 

108. 
104- 

135. 
181. 

190. 
151. 

12th  in  speed 

142 

(42) 

Fixed 

104. 

108. 

125. 

167. 

183. 

143 

Same  figures  as  for  43 

144 

Same  figures  as  for  44 

145 

Same  figures  as  for  43 

146 

Same  figures  as  for  44 

*  As  revised  in  Chapter  XVI,  8  9. 


APPENDIX  VII 


507 


TABLE  65  (Continued) 
AGGREGATIVE  CROSSES 


RANKS*  OF  FIRST 

20  IN  ACCURACY, 

IDENTI- 

SPEED, SIMPLIC- 

FICATION 

BASE 

1914 

1915 

1916 

1917 

1918 

ITY  OP  FORMULA, 

NUMBER 

AND  CONFORMITY 

TO  CIRCULAR 

TEST 

151 

Fixed 

95.88 

96.29 

107.70 

146.90 

172.76 

1st  in  speed 

(51) 

Chain 

Same  as  Fixed  Base  (necessarily) 

1st  in  sim- 

plicity 

1st  in  con- 

formity 

152 

Fixed 

97.12 

97.18 

114.55 

158.65 

165.15 

5th  in  speed 

(52) 

Chain 

Same  as  Fixed  Base  (necessarily) 

20th  in  sim- 

plicity 

1st  in  con- 

formity 

153 

Same  as  353  (necessarily) 

154 

Same  as  353  (necessarily) 

*  As  revised  in  Chapter  XVI,  §  9. 

CROSS  FORMULA  (200-299) 

Those  for  which  figures  are  given  conform  to  Test  2  only. 
ARITHMETIC  CROSSES 


IDENTI- 
FICATION 

NUMBER 

201 

BASE 

1914 

1918 

1916 

1917 

1918 

R,ANKS*  OF  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

Fixed 

98.23 

96-97 

116.43 

163.87 

181.85 

203 

Same  as  353  (necessarily) 

205 

Same  as  353  (necessarily) 

207 

Fixed 

99.78 

99.54 

114.56 

165.88 

176.74 

209 

Fixed 

99.81 

99.61 

114.63 

167.35 

178.00 

*  As  revised  in  Chapter  XVI,  §  9. 


508 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  65  (Continued) 
HARMONIC  CROSSES 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OP  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

211 

Fixed 

99.24 

98.40 

117.22 

164.84 

204.60 

213 

Fixed 

100.53 

100.10 

113.87 

157.42 

178.88 

215 

Fixed 

100.49 

100.03 

113.79 

156.34 

177.76 

217 

Same  as  353  (necessarily) 

219 

Same  as  353  (necessarily) 

*  As  revised  in  Chapter  XVI,  §  9. 


GEOMETRIC  CROSSES 


IDENTI- 
FICATION 

NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,    SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

221 

Same  as  321  (necessarily) 

223 

Fixed 

100.31 
100.29 

100.01 

113.52 

159.37 

178.01 

225 

Fixed 

99.97 

113.99 

158.78 

177.42 

227 
229 

Fixed 

99.95 

99.74 

114.39 

163.81 

177.29 

Fixed 

99.96 

99.78 

114.45 

164.47 

177.95 

*  As  revised  in  Chapter  XVI,  {  9. 


APPENDIX  VII 


509 


TABLE  65  (Continued) 
MEDIAN  CROSSES 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OP  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

231 

Same  as  331  (necessarily) 

233 

Fixed 

100.77 

101.99 

112.27 

160.76 

175.44 

235 

Fixed 

100.72 

101.69 

112.38 

162.81 

175.53 

237 

Fixed 

100.59 

100.85 

112.69 
112.90 

160.81 

183.94 

239 

Fixed 

100.63 

100.87 

161.86 

179.22 

*  A3  revised  in  Chapter  XVI,  §  9. 


MODE  CROSSES 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

241 

Same  as  341  (necessarily) 

243 

Fixed 

102. 

103. 

119. 

179. 

174. 

245 

Same  figures  as  for  243 

247 

Same  figures  as  for  243 

249 

Same  figures  as  for  243 

*  As  revised  in  Chapter  XVI,  §  9. 


510 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  65  (Continued) 
AGGREGATIVE  CROSSES 


IDENTI-  ' 

FICATION 

NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OF  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

251 

Same  as  351  (necessarily) 

253 

Same  as  353  (necessarily) 

259 

Same  as  353  (necessarily) 

*  As  revised  in  Chapter  XVI,  §  9. 


CROSS  FORMULAE   (300-399) 

Fulfilling  both  tests 
ARITHMETIC  AND  HARMONIC  CROSSES 


IDENTI- 
FICATION 
NUMBER 

BASK 

1914 

1915 

1916 

1917 

1918 

RANKS*  OF  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

301 

Fixed 

98.73 

97.68 

116.82 

164.35 

192.89 

303 

Same  as  353  (necessarily) 

305 

Same  as  353  (necessarily) 

307 

Fixed 

100.13 

99.78 

114.17 

161.04 

177.25 

309 

Fixed 

100.17 

99.85 

114.25 

162.31 

178.44 

*  As  revised  in  Chapter  XVI,  5  9. 


APPENDIX  VII 


511 


TABLE  65  (Continued) 
GEOMETRIC  CROSSES 


IDENTI- 
FICATION 

NUMBER 

BASK 

1914 

1915 

1916 

1917 

1918 

t,ANK8*  OP  FIRST 
0  IN  ACCURACY, 

PEED,    SlMPLIC- 

TY  OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

321 

(221) 

Fixed 
Chain 

98.70 
Sam 

97.70 
B  as  Fixe 

116.91 
d  Base  ( 

164.25 
necessari 

187.00 

ly) 

1st  in  con- 
formity 

323 

Fixed 
Chain 

100.13 
100.13 

99.89 
100.23 

113.99 
114.45 

161.90 
162.47 

177.98 
178.69 

9th  in  ac- 
curacy 
10th  in  con- 
formity 

325 

Fixed 
Chain 

100.12 
100.12 

99.85 
100.23 

114.19 
114.45 

161.28 
162.36 

177.35 
178.58 

8th  in  ac- 
curacy 
9th  in  con- 
formity 

*  As  revised  in  Chapter  XVI,  §  9. 


MEDIAN  CROSSES 


IDENTI- 
FICATION 

NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OF  FIRST 
20  IN  ACCURACY, 
SPEED,    SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

331 
(231) 

333 
335 

Fixed 

99.78 

100.37 

117.40 

162.98 

187.19 

Fixed 

100.70 

101.43 

112.59 

161.31 

177.32 

Fixed 

100.65 

101.27 

112.53 

161.81 

179.69 

*  Aa  revised  in  Chapter  XVI,  §  9. 


512 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  65  (Continued} 
MODE  CROSSES 


RANKS*  OF  FIRST 

20  IN  ACCURACY, 

IDENTI- 

SPEED, SIMPLIC- 

FICATION 

BASE 

1914 

1915 

1916 

1917 

1918 

ITY  OF  FORMULA, 

NUMBER 

AND  CONFORMITY 

TO  CIRCULAR 

TEST 

341 

Fixed 

100.96 

102.88 

116.19 

150.15 

186.47 

(241) 

343 

Same  figures  as  for  243 

345 

Same  figures  as  for  243 

*  Aa  revised  in  Chapter  XVI,  §  9. 


AGGREGATIVE  CROSSES 


RANKS*  OF  FIRST 

20  IN  ACCURACY, 

IDENTI- 

SPEED, SIMPLIC- 

FICATION 

BASE 

1914 

1915 

1916 

1917 

1918 

ITY  OF  FORMULA, 

NUMBER 

AND  CONFORMITY 

TO  CIRCULAR 

TEST 

351 

Fixed 

96.50 

96.73 

111.07 

152.66 

168.91 

llth  in  speed 

(251) 

Same  as  Fixed  Base  (necessarily) 

1st  in  con- 

formity 

353  f 

Fixed 

100.12 

99.89 

114.21 

161.56 

177.65 

1st  in  ac- 

curacy 

Chain 

100.12 

100.23 

114.32 

162.23 

178.49 

17th  in  sim- 

plicity 

2nd  in  con- 

formity 

*  As  revised  in  Chapter  XVI,  §  9. 

t  353  -  103,  104,  105,  106,  153,  154,  203,  205,  217,  219,  253,  259,  303,  305. 


APPENDIX  VII 


513 


TABLE  65  (Continued) 

CROSS  WEIGHT  FORMULAE   (1000-4999) 
CROSS  WEIGHT  ARITHMETIC  AND  HARMONIC 
1000-1099  fulfill  neither  test. 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OP  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

1003 

Fixed 

100.45 

100.93 

116.02 

170.81 

182.54 

1004 

Fixed 

99.47 

98.60 

112.84 

158.01 

173.03 

1013 

Fixed 

99.81 

98.91 

112.53 

153.51 

173.02 

1014 

Fixed 

100.83 

101.10 

115.54 

165.24 

182.94 

CROSSES  OF  PRECEDING 
1100-1199  fulfill  Test  1  only. 


1103 

Fixed 

100.13 

99.91 

114.26 

161.93 

177.72 

1104 

Fixed 

100.15 

99.84 

114.18 

161.58 

177.92 

*  As  revised  in  Chapter  XVI,  §  9. 


514 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  65  (Continued) 
CROSS  WEIGHT  GEOMETRIC,  MEDIAN,  MODE,  AGGREGATIVE 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

R.ANKS*  OP  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

1123 

Fixed 
Chain 

100.14 

100.14 

99.89 

100.24 

114.17 
114.94 

161.62 

162.06 

177.87 
178.40 

18th  in  ac- 
curacy 

1124 

Fixed 
Chain 

100.12 
100.12 

99.91 
100.24 

114.28 
115.05 

161.78 
163.36 

177.73 

179.70 

19th  in  ac- 
curacy 

1133 

Fixed 

100.52 

99.57 

108.39 

162.63 

170.85 

1134 

Fixed 

100.75 

103.33 

117.53 

162.59 

182.15 

1143 

Same  figures  as  for  43 

1144 

Same  figures  as  for  44 

1153 

Fixed 
Chain 

100.13 
100.18 

99.89 
100.23 

114.20 
114.30 

161.70 

162.21 

177.83 
178.37 

12th  in  ac- 
curacy 
14th  in  sim- 
plicity 

1154 

Fixed 

100.12 

99.90 

114.24 

161.73 

177.76 

13th  in  ac- 
curacy 

*  Aa  revised  in  Chapter  XVI,  §  9. 


APPENDIX  VII 


515 


TABLE  65  (Continued) 

CROSSES  OR  CROSS  WEIGHT  FORMULAE,  ALL  TYPES  (1300-1399) 
Fulfilling  both  tests 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1916 

1917 

1918 

RANKS*  OF  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

1303 

Fixed 

100.14 

99.88 

114.22 

161.75 

177.82 

1323 

Fixed 
Chain 

100.13 
100.18 

99.90 
100.24 

114.23 
114.65 

161.70 

162.71 

177.80 
179.05 

5th  in  ac- 
curacy 
6th    in    con- 
formity 

1333 

Fixed 

100.63 

101.43 

112.87 

162.61 

176.41 

1343 

Same  figures  as  for  243 

1353 

Fixed 
Chain 

100.13 
100.13 

99.89 
100.23 

114.22 
114.88 

161.71 

162.27 

177.79 

178.45 

4th  in  ac- 
curacy 
5th    in    con- 
formity 

*  As  revised  in  Chapter  XVI,  $  9. 


516 


THE  MAKING  OF  INDEX  NUMBERS 


TABLE  65  (Continued) 

OTHER  CROSS  WEIGHT  FORMULA  (2000-4999) 
2100-2199 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1916 

1916 

1917 

1918 

RANKS*  OF  FIRST 
20  IN  ACCURACY, 
SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

2153 

Fixed 
Chain 

100.12 
100.12 

99.89 
1  00.23 

99.90 

100.24 

114.23 
114*4 

161.52 

162.25 

177.63 

178.52 

10th  in  ac- 
curacy 
14th  in  speed 
8th  in  sim- 
plicity 
llth  in  con- 
formity 

2154 

Fixed 
Chain 

100.14 
100.14 

114.21 
114.31 

161.69 

162.38 

177.72 
178.65 

llth  in  ac- 
curacy 

2300-2399 

2353 

Fixed 
Chain 

100.13 
100.13 

99.89 
100.23 

114.22 

114.32 

161.60 
162.31 

177.67 

178.58 

2nd  in  ac- 
curacy 
3rd  in  con- 
formity 

3100-3199 

3153 

Fixed 

100.15 

99.88 

114.23 

162.11 

176.94 

3154 

Fixed 

100.12 

99.92 

114.28 

161.77 

177.78 

3300-3399 

3353 

Fixed 
Chain 

100.14 
100.14 

99.90 

100.24 

114.35 

114.28 

161.94 
162.14 

177.36 
178.39 

20th  in  ac- 
curacy 

4100-4199 

4153 

Fixed 
Chain 

100.12 
100.12 

99.97 
100.25 

114.44 
114.55 

162.40 
162.45 

178.26 
178.79 

4154 

Fixed 
Chain 

100.14 
100.14 

99.88 
100.24 

114.08 
114.20 

161.16 
161.96 

176.79 
178.14 

4300-4399 

4353 

Fixed 

100.13 

99.92 

114.26 

161.78 

177.52 

*  As  revised  in  Chapter  XVI,  §  9. 


APPENDIX  VII 


517 


TABLE  65  (Continued} 

MISCELLANEOUS  FORMULA   (5000-9999) 
CROSSES  OP  CROSS  FORMULA  (5000-5999) 


IDENTI- 
FICATION 
NUMBER 

BASE 

1914 

1915 

1913 

1917 

1918 

RANKS*  OF  FIRST 

20   INACCURACY, 

SPEED,  SIMPLIC- 
ITY OF  FORMULA, 
AND  CONFORMITY 
TO  CIRCULAR 
TEST 

5307 

Fixed 

100.15 

99.82 

114.21 

161.67 

177.84 

5323 

Fixed 
Chain 

100.13 
100.13 

99.87 
100.23 

114.09 
114.45 

161.59 
162.42 

177.67 
178.64 

3rd  in  ac- 
curacy 
4th  in  con- 
formity 

5333 

Fixed 

100.68 

101.35 

112.56 

161.56 

178.50 

5343 

Same  figures  as  for  243 

BROADENED  BASE  FORMULA  (6000-6999) 


6023 
('13-'14) 

100.12 

99.50 

112.25 

153.53 

173.45 

19th  in  sim- 
plicity 
1st  in  con- 
formity 

6023 
('13-'16) 

99.93 

99.88 

113.61 

156.61 

175.32 

ditto 

6023 
('13  &  '18) 

99.45 

99.12 

114.23 

159.93 

179.54 

ditto 

6053 
C13-14) 

100.12 

100.09 

113.89 

161.26 

177.73 

7th  in  speed 
7th  in  sim- 
plicity 
1st  in   con- 
formity 

6053 
('13-'16) 

100.02 

100.04 

113.99 

161.88 

178.24 

ditto 

6053 
('13-'18) 

99.79 

99.85 

114.04 

161.59 

177.88 

ditto 

*As  revised  in  Chapter  XVI,  §  9. 


518         THE  MAKING  OF  INDEX  NUMBERS 
TABLE  65  (Continued) 


RANKS*  OP  FIRST 

20  IN  ACCURACY. 

IDENTI- 

SPEED, SIMPLIC- 

FICATION 

BASE 

1914 

1915 

1916 

1917 

1918 

ITY  OF  FORMULA, 

NUMBER 

AND  CONFORMITY 

TO  CIRCULAR 

TEST 

AVERAGE  OF  353  BY  Six  BASES  (7000-7999) 


7053  | 

100.09 

99.96 

114.03 

161.53 

177.90 

ARITHMETIC  AND  HARMONIC  MEANS  OF  AGGREGATIVES  (8000-8999) 


S053 

Fixed 

100.12 

99.89 

114.21 

161.56 

177.65 

6th  in  ac- 

Chain 

100.12 

100.23 

114.33 

162.24 

178.50 

curacy 
15th  in  sim- 
plicity 
7th  in  con- 
formity 

8054 

Fixed 

100.12 

99.89 

114.21 

161.56 

177.65 

7th  in  ac- 

Chain 

100.12 

100.23 

114-32 

162.23 

178.49 

curacy 
16th  in  sim- 
plicity 
8th  in  con- 
formity 

8353 

(cross  of  above)  =  353 

ROUND  WEIGHT  FORMULAE  (9000-9999) 


QOOlf 

llth  in  sim- 
plicity 

9011f 

12th  in  sim- 
plicity 

9021f 

13th  in  sim- 
plicity 
1st  in  con- 
formity 

9051 

Fixed 

101.68 

103.10 

113.63 

160.37 

182.07 

2nd  in  speed 
2nd  in  sim- 
plicity 
1st  in  con- 
formity 

*  As  revised  in  Chapter  XVI,  §  9.         f  Not  calculated.     See  footnote  to  Table  47,  p.  348. 


APPENDIX  VIII 

SELECTED  BIBLIOGRAPHY 

1863.  William  Stanley  Jevons.  Investigations  in  Our  Currency  and 
Finance.  Sections  II-IV,  pp.  13-150.  London,  1909.  (Reprints  of 
various  articles  published  in  1863,  etc.) 

1887-1889.  F.  Y.  Edgeworth.  Reports  of  the  Committee  (of  the  British 
Association  for  the  Advancement  of  Science)  appointed  for  the  purpose 
of  investigating  the  best  methods  of  ascertaining  and  measuring  va- 
riations in  the  value  of  the  monetary  standard.  In  Reports  of  the  As- 
sociation published  in  1888,  pp.  254-301;  1889,  pp.  188-219;  1890, 
pp.  133-64. 

1901.  Correa  Moylan  Walsh.  The  Measurement  of  General  Exchange- 
Value.  580  pp.  Macmillan,  1901. 

1903.  H.  Fountain.  "  Memorandum  on  the  Construction  of  Index  Num- 
bers of  Prices,"  from  Report  on  Wholesale  and  Retail  Prices  in  the 
United  Kingdom  in  1902,  House  of  Commons  Paper  No.  321  of  1903, 
pp.  429-52.  Darling  &  Son,  1903. 

1911.  Irving  Fisher.     The   Purchasing   Power  of  Money,   pp.   198-234, 
pp.  385-430.     Macmillan,  1911. 

1912.  G.  H.  Knibbs.     Prices,  Price  Indexes,  and  Cost  of  Living  in  Aus- 
tralia.   Commonwealth  Bureau  of  Census  and  Statistics,  Labour  and 
Industrial  Branch,  Report  No.  1,  Appendix.    McCarron,  Bird  &  Co., 
Melbourne,  December,  1912. 

1915.  Wesley  C.  Mitchell.     Index   Numbers  of  Wholesale   Prices  in  the 
United  States  and  Foreign  Countries.     U.  S.  Bureau  of  Labor  Statistics, 
Bulletin  284,  October,  1921.     (Revision  of  Bulletin  173,  July,  1915). 

1916.  Frederick  R.  Macaulay.     "  Making  and  Using  of  Index  Numbers." 
American  Economic  Review,  pp.  203-9,  March,  1916. 

1916.  Wesley  C.  Mitchell.  "  A  Critique  of  Index  Numbers  of  the  Prices 
of  Stocks."  Journal  of  Political  Economy,  pp.  625-93,  July,  1916. 

1918.  G.  H.  Knibbs.     Price  Indexes,   Their  Nature  and  Limitations,  the 
Technique  of  Computing  Them,  and  Their   Application  in  Ascertain- 
ing  the    Purchasing    Power   of   Money.     Commonwealth  Bureau  of 
Census  and  Statistics,  Labour  and  Industrial  Branch,  Report  No.  9, 
Appendix.     McCarron,  Bird  &  Co.,  Melbourne,  1918. 

1919.  A.  L.  Bowley.     "The  Measurement  of  Changes  in  the  Cost  of 
Living."     Journal  of  the  Royal  Statistical  Society,  pp.  343-61,  May, 
1919. 

1920.  A.  C.  Pigou.     The  Economics  of  Welfare,  pp.  69-90.     Macmillan, 
1920. 

1921.  G.  E.  Barnett.     "  Index  Numbers  of  the  Total  Cost  of  Living." 
Quarterly  Journal  of  Economics,  pp.  240-63,  February,  1921. 

519 


520         THE  MAKING  OF  INDEX  NUMBERS 

1921.     Irving  Fisher.     "  The  Best  Form  of  Index  Number."     Quarterly 

Publication  of  the  American  Statistical  Association,  pp.  533-51,  March, 

1921. 
1921.     A.  W.  Flux.     "  The  Measurement  of  Price  Changes."  Journal  of 

the  Royal  Statistical  Society,  pp.  167-215,  March,  1921. 
1921.     Correa  Moylan  Walsh.     The   Problem  of  Estimation.     139  pp. 

P.  S.  King,  London,  1921. 
1921.     Warren  M.  Persons.     "  Fisher's  Formula  for  Index  Numbers." 

Review  of  Economic  Statistics,  pp.  103-13,  May,  1921. 
1921.     Allyn  A.  Young.     "  The  Measurement  of  Changes  of  the  General 

Price  Level."    Quarterly  Journal  of  Economics,  pp.  557-73,  August, 

1921. 
1921.     Truman  L.   Kelley.     "  Certain  Properties  of  Index  Numbers." 

Quarterly   Publication    of  the    American  Statistical   Association,   pp. 

826-41,  September,  1921. 
1921.     Lucien  March.     "  Les  modes  de  mesure  du  mouvement  ge"ne"ral 

des  prix."     Metron,  pp.  57-91,  September,  1921. 

(For  completer  references  see  the  bibliographies  issued  from  time  to 
time  by  the  Library  of  Congress.) 


INDEX 


References  to  pages  where  technical  terms  are  defined  or  explained,  have 
been  set  in  boldface  type. 

Aberthaw  Index,  cited,  368. 
Accuracy  of  index  numbers,  330-349. 
Aggregative,  the  word,  15,  371;  fixed 

base   and   chain  methods  agree   for 

simple,  373. 
Aggregative    average,    simple,    39-40; 

peculiarities  of  the,  378-379. 
Aggregative      formulae,      systems      of 

weighting  for,  56-57;   cross  weight, 

187;    list    of,    201;    comments    on, 

234-237. 

Aldrich  Senate  Report,  cited,  333,  445. 
Alexander  Hamilton   Institute,   cited, 

438. 

Algebraic  notations,  key  to,  461. 
American  Institute  of  Finance,  cited, 

438. 
American   Writing   Paper   Co.   Index, 

cited,  368. 
Antitheses,     rectifying     formulae     by 

crossing    time,    136-142;    rectifying 

formulae  by  crossing  factor,  142-144, 

396-397;    fourfold    relationship    of, 

144-145. 
Antithesis,    time,     118;    factor,     118; 

numerical  and  graphic  illustrations 

of    time,    119-120;    numerical    and 

graphic  illustrations  of  factor,  125- 

130. 
Arithmetic    average,    simple,     15-23; 

among  the  worst  of  index  numbers, 

29-30;   lies   above   geometric,   375- 

377. 
Arithmetic     formulae,     cross     weight 

harmonic     and,     187-189;     list     of 

harmonic  and,  199. 
Arithmetic     forward     by     arithmetic 

backward   exceeds   unity,    383-384. 
Attributes  of  index  number,  8-9. 
Australian    Bureau    of    Census    and 

Statistics,  cited,  363. 
Average,  index  number  defined  as  an, 

3;  a  simple,  4-6;  a  weighted,  6-8; 

note  on  definition  of  word,  373-375. 

See  under  Aggregative,  Arithmetic, 


Geometric,  Harmonic,   Median,  and 
Mode. 

Averaging,  136;  of  various  individual 
quotations  for  one  commodity, 
317-318. 

Babson,  cited,  438,  460. 

Barnett,  G.  E.,  cited,  519. 

Base,  fixed,  15-18;  chain,  18-22. 

Base  number,  18,  371. 

Base  year,  19. 

Base  year  values,  weighting  by, 
compared  with  weighting  by  given 
year  values,  45-53. 

Bell,  Charles  A.,  statement  by,  on 
method  of  splicing  employed  by 
U.S.  Bureau  of  Labor  Statistics, 
427-428. 

Bias,  86;  single,  86;  in  arithmetic  and 
harmonic  types  of  formulae,  86-88; 
weight  and  type,  91-94;  double,  102- 
105;  relation  between  dispersion  and, 
108-111,  387-395;  errors  and,  gener- 
ally relative,  116-117;  use  of  term  by 
different  statisticians,  117;  formulae 
characterized  by,  capable  of  rectifi- 
cation, 266;  tables  of  deviation  and, 
390,  392,  393;  of  Formulae  53  and  54 
slight,  410-412;  of  6023  and  23  as 
affected  by  price-quantity  corre- 
lation, 428;  more  disturbing  than 
chance,  in  weighting,  446-447. 

Bibliography  on  index  numbers,  519- 
520. 

Blending,  305;  substitutes  for,  306- 
308. 

Bowley,  A.  L.,  simple  median  average 
approved  by,  36;  use  of  term  "bias" 
by,  117;  cited,  519. 

Bradstreet,  cited,  207,  333,  460;  simple 
aggregative  approved  by,  459,  471. 

British  Board  of  Trade,  cited,  332,  333, 
438. 

British  Imperial  Statistical  Confer- 
ence, resolution  passed  by,  on 


522 


INDEX 


methods  of  constructing  index  num- 
bers, 240-241. 

Broadened  base  system,  312-313. 

Brookmire,  cited,  438. 

Burchard,  H.  C.,  index  number  con- 
structed by,  459. 

Calculation  of  formulae,  speed  of,  321- 
329. 

Calculation  of  weighted  median  and 
mode,  377-378. 

Canadian  Department  of  Labor,  cited, 
332,  334. 

Carli,  G.  R.,  simple  arithmetic  average 
approved  by,  29,  458,  466. 

Chain  base  system,  18-22;  for  simple 
geometric  fixed  base  system  agrees 
with,  371-372 ;  for  simple  aggregative 
fixed  base  system  agrees  with,  373. 

Circular  test,  270-271;  illustration  of 
non-fulfillment,  by  case  of  three 
unlike  countries,  271-272;  can  be 
fulfilled  only  if  weights  are  constant, 
274-276;  question  as  to  how  near  to 
fulfillment  in  actual  cases,  276  ff.; 
the  circular  gap,  or  deviation  from 
fulfilling  circular  test  of  Formula  353, 
278-288;  status  of  all  formulas  rela- 

'  tively  to,  288-292;  reduction  of,  to  a 
triangular  test,  295;  note  on  alge- 
braic expression  of,  413;  conforma- 
tion of  simple  or  constant  weighted 
geometric  to,  413,  416;  formula 
satisfying,  for  three  dates  will  satisfy 
for  four,  etc.,  426-^27. 

Circular  (test)  gap,  277-280;  tabu- 
lation of,  for  Formula  353,  280-283; 
discussion  of,  of  Formula  353,  283- 
287;  comparison  of,  of  134  different 
formulae,  287-288;  meaning  of 
"equal  and  opposite,"  418. 

Coggeshall,  F.,  harmonic  index  number 
approved  by,  33,  467. 

Commensurability,  as  test  of  index 
number  of  prices,  420-426. 

Commodity  reversal  tests,  63-64. 
See  under  Tests. 

Cross  between  two  factor  antitheses 
fulfills  Test  2,  396-397. 

Cross  formula,  185,  407. 

Cross  references  between  "Purchasing 
Power  of  Money"  and  this  book, 
419. 

Cross  weight  formula,  185. 

Crossing  of  formulae,  136-183. 


Crossing  of  weights  possible  geometri- 
cally, arithmetically,  harmonically, 
401-407. 

Davies,  George  R.,  Formula  353 
approved  by,  242. 

Day,  E.  E.,  studies  by,  14  n.;  quantity 
figures  worked  out  by,  110;  Formula 
6023  approved  by,  486;  cited,  253, 
254,  313,  314,  316,  317,  326,  328, 
342,  343,  384. 

Determinateness,  as  test  of  index  num- 
ber of  prices,  420-423. 

Deviation,  standard,  no;  tables  of, 
337,  391;  tables  of  bias  and,  390, 
392,  393. 

Dispersion,  108;  dependence  of  bias  on, 
108-111;  charts  showing,  measured 
by  standard  deviations,  290-294; 
notes  on  bias  and,  in  formulae,  387- 
390;  "skewness"  of,  408-410. 

Dispersion  index,  tables  showing, 
compared  with  standard  deviation, 
392,  393. 

Drobisch,  M.  W.,  use  of  factor  an- 
titheses by,  134;  cross  formula  sug- 
gested by,  196;  Formula  52  approved 
by,  471;  Formula  8053  approved  by, 
487. 

Dun,  Formula  53  approved  by,  471; 
Formula  9001  approved  by,  487; 
cited,  336,  460. 

Dutot,  simple  aggregative  index  num- 
ber approved  by,  40,  458,  471. 

Economist  (London),  simple  arithmetic 
approved  by,  459,  466;  cited,  29,  333. 

Edgeworth,  F.  Y.,  simple  median 
approved  by,  36,  262,  469;  cross 
weight  aggregative  proposed  by, 
196;  "probability"  system  of 
weighting  of,  379-380;  Formula 
2153  approved  by,  484;  recommen- 
dations of,  with  regard  to  index 
numbers,  459;  cited,  255,  320,  365, 
366,  408,  519. 

Entry,  as  test  of  index  number  of 
prices,  420-423. 

Erratic  index  numbers,  112-116. 

Errors,  joint.   See  Joint  errors. 

Errors,  probable.    See  Probable  error. 

Factor  antithesis.   See  Antithesis. 
Factor  reversal  tests,   72.    See  under 
Tests. 


INDEX 


523 


Fairness,  a  requirement  in  index 
numbers,  9,  10,  62. 

Falkner,  R.  P.,  Formula  9001  ap- 
proved by,  459,  487. 

Federal  Reserve  Board,  geometric 
weighted  by  given  year  values  ap- 
proved by,  468;  cited,  460. 

Fisher,  Irving,  cited,  25,  82,  242,  520; 
The  Rate  of  Interest,  cited,  63  n.; 
Purchasing  Power  of  Money,  cited, 
82,  381,  519;  relation  of  present  book 
to  Appendix  on  Index  Numbers  in 
Purchasing  Power  of  Money,  418- 
426;  Formula  53  approved  by,  471; 
Formula  54  approved  by,  471;  ideal 
index  number  approved  by,  482; 
Formula  2153  approved  by,  484. 

Fisher,  Willard,  cited,  458. 

Fixed  base  system,  15-18;  for  simple 
geometric  chain  system  agrees  with, 
371-372;  for  simple  aggregative  chain 
system  agrees  with,  373. 

Flux,  A.  W.,  cited,  111,  296,  320,  366, 
520;  simple  geometric  approved  by, 
468. 

Formulae,  classification  of,  in  six  types, 
15;  time  reversal  tests  as  finders  of, 
118-135;  rectifying,  by  crossing 
them,  136-183;  rectifying,  by 
crossing  their  weights,  184-196; 
lists  of,  170-174;  main  series  of, 
184,  197;  supplementary  series  of, 
184;  seven  classes  of,  202;  compari- 
son of,  with  view  to  selecting  the 
best,  206-212;  comparison  of  other, 
with  the  "ideal"  (Formula  353), 
243-269;  eight  most  practical,  361- 
362;  method  for  comparing  with 
"ideal,"  412-413;  key  to  numbering 
of,  461-465;  table  of,  for  index  num- 
bers, 466-^88;  alternative  forms  of 
certain,  488. 

Fountain,  H.,  cited,  519. 

Freakishness,  of  median  and  mode, 
112-116,  209-211;  of  simple  aggre- 
gative, 207-209;  lessening,  by  in- 
creasing number  of  commodities, 
216-218;  formula  rendered  wholly 
unreliable  by,  266. 

Geometric  average,  simple,  33-35; 
cross  weight,  186;  list  of  formula}, 
200;  comparison  of  the  simple,  and 
the  simple  median,  260-264;  fixed 
base  and  chain  methods  agree  for, 


371-372;    lies    between    arithmetic 

above  and  harmonic  below,  375-377. 
Gibson,  Thomas,  cited,  460. 
Given    year    values,     weighting    by, 

compared   with   weighting  by   base 

year  values,  45-53. 

Haphazard,  applied  to  weighting,  207; 
index  numbers  found  to  be,  218; 
differences  between  simple  and  cross 
weight  index  numbers  are,  444.  See 
Freakishness. 

Harmonic  average,  30;  the  simple,  30- 
33;  lies  below  geometric,  375-377. 

Harmonic  formulae,  cross  weight  arith- 
metic and,  187-191;  list  of,  199. 

Harvard  Committee  on  Economic 
Research,  cited,  53,  438,  460. 

Hastings,  Hudson,  cited,  429. 

Historical  notes,  on  methods  of 
weighting,  59-60;  on  reversal  tests, 
82;  on  biased  index  numbers,  117; 
on  tests  as  finders  of  formulae,  134- 
135;  on  crossing  of  formulae,  183;  on 
crossing  of  weights,  196;  on  Formula 
353,  240-242;  on  circular  test,  295- 
296;  on  fixed  base,  broadened  base, 
and  chain  systems,  320;  on  USP  of 
index  numbers,  458-460. 

Hofmann,  Emil,  cited,  437. 

Holt  &  Co.    Index,  cited,  368. 

Hybrid  weighting,  53-56. 

Ideal  blend,  305-306. 

Ideal  index  number  (Formula  353), 
220-225;  probable  error  of,  225-229; 
history  of,  240-242;  Formula  2153 
close  to,  428-430. 

Index  numbers,  3;  simple  arithmetic, 
4-5;  weighted  arithmetic,  6-7; 
attributes  of,  8-9;  fairness  of,  9-10, 
62;  six  types  of,  compared,  11  ff. ; 
simple  harmonic,  30-33;  simple 
geometric,  33-35;  simple  median, 
35-36;  simple  mode,  36-39;  simple 
aggregative,  39-40;  comparison  of 
six  simple  forms  of,  41-42;  calcu- 
lation of,  by  different  methods  of 
weighting,  43-56;  only  two  systems 
of  weighting  for  aggregative  type  of, 
56-57;  relation  of  weighted  aggre- 
gative to  weighted  arithmetic  and 
weighted  harmonic,  60,  379;  reversal 
tests  of,  62-82 ;  joint  errors  between, 
83-86;  erratic  and  freakish,  112-116; 


524 


INDEX 


rectification~of  formulae,  by 'crossing, 
136-183 ;  rectifying  formulae  by  cross- 
ing their  weights,  184-196;  the  best 
simple,  206-212;  finding  the  very 
best,  213-242;  comparison  of  all, 
with  Formula  353,  243-269;  results 
of  comparisons  among  134  varieties, 
266-269;  so-called  circular  test  of, 
270-296;  blending  apparently  in- 
consistent results,  297-320;  influ- 
ence of  assortment  and  number  of 
samples,  331-340;  future  uses  of, 
367-369;  list  of  discontinued,  432- 
433;  list  of  current,  433^38;  in- 
fluence of  weighting  on,  439-457; 
averages  of  ratios  rather  than  ratio 
of  averages,  451-457;  landmarks  in 
history  of,  458-460;  list  of  formulas 
for,  462-488;  examples  showing  how 
to  calculate,  490-497;  tables  of, 
by  134  formulas,  498-518;  bibliog- 
raphy on  subject  of,  519-520. 

Institute  of  Finance,  American,  cited, 
438. 

International  Labour  Office,  cited,  438. 

International  Labour  Review,  cited,  437. 

Jevons,  W.  S.,  simple  geometric  ap- 
proved by,  35,  459,  468;  cited,  139, 
296,  519. 

Joint  errors  between  index  numbers, 
83-86;  expressible  by  product  or 
quotient,  88-90. 

Kelley,  Truman  L.,  "Certain  Proper- 
ties of  Index  Numbers"  by,  cited, 
331,  334,  340,  424,  520;  method 
proposed  by,  of  measuring  probable 
error  of  index  number,  430-431. 

Kemmerer,  E.  W.,  cited,  368. 

Key,  to  principal  algebraic  notations, 
461;  to  numbering  of  formulae  of 
index  numbers,  461-465. 

Knibbs,  G.  H.,  weighted  aggregative 
formula  approved  by,  59,  240,  460, 
471;  cited,  230,  366,  371,  519. 

Laspeyres,  E.,  weighted  aggregative 
formula  approved  by,  59,  459,  471; 
formula  of,  in  relation  to  factor 
antithesis,  131-132;  cited,  60,  161, 
168,  169,  240,  255,  320,  387,  412. 

Laughlin,  J.  L.,  cited,  458. 

Lehr,  J.,  cited,  134,  196,  255,  326; 
Formula  2154  approved  by,  485. 


Linking,  process  of,  22. 

London   School  of  Economics,   cited, 

438. 
Lowe,  Joseph,  Formula  9051  approved 

by,  458,  487. 

Macalister,  "Law  of  the  Geometric 
Mean,"  cited,  231  n. 

Macaulay,  F.  R.,  theorem  of,  relative 
to  so-called  circular  test,  292-293; 
cited,  241,  366,  426,  519. 

Main  series  of  formulae,  184,  197. 

March,  Lucien,  cited,  296,  520. 

Marshall,  Alfred,  cross  weight  aggre- 
gative approved  by,  196,  484;  chain 
base  system  suggested  by,  320; 
cited,  255,  365,  366. 

Massachusetts  Commission  on  the 
Necessaries  of  Life,  cited,  460. 

Median  average,  simple,  35~36;  freak- 
ishness  of,  210-211;  compared  with 
simple  geometric,  260-264;  calcu- 
lation of  weighted,  377-378. 

Median  formulae,  cross  weight,  186; 
list  of,  200;  comments  on,  258-260. 

Meeker,  Royal,  cited,  240,  366. 

Messedaglia,  A.,  cited,  459. 

Method  for  comparing  other  formulae 
with  "ideal,"  412^13. 

Mitchell,  Wesley  C.,  data  collected  by, 
14;  use  of  simple  median  average  by, 
36,  469;  cited,  38,  39,  216,  232,  233, 
295,  331,  332,  334,  335,  336,  366, 
371,  408,  426,  445,  458,  460,  519. 

Mode,  simple,  36-39;  method  of 
finding  the  simple,  372-373;  calcula- 
tion of  weighted,  377-378;  if  above 
geometric  forward,  below  it  back- 
ward, 407. 

Modes,  cross  weight,  186;  list  of  for- 
mulas in  group  of,  201;  comments 
on,  258-260. 

National  Industrial  Conference  Board, 

cited,  438,  460. 
Neumann-Spallart,  cited,  438. 
Nicholson,  J.  S.,  cited,  134;  Formula 

22  approved  by,  468. 
Numbering    of    formulae,    system    of, 

142,  461-465. 

Ogburn,  W.  F.,  theorem  of,  relative  to 
so-called  circular  test,  292-293; 
formula  of,  for  Macaulay's  Theorem, 
426. 


INDEX 


525 


Paasche,  H.,  weighted  aggregative 
formula  of,  60;  formula  of,  in  relation 
to  factor  antithesis,  131-132;  For- 
mula 54  approved  by,  459,  471; 
cited,  161,  168,  169,  240,  255,  387. 

Palgrave,  R.  H.  Inglis,  arithmetic 
weighted  by  given  year  values  ap- 
proved by,  466;  cited,  102,  103,  111. 

Pearl,  Raymond,  cited,  382. 

Percentaging,  16. 

Persons,  W.  M.,  studies  by,  14  n.; 
quantity  figures  worked  out  by, 
110;  index  of  crops,  236-239;  refer- 
ence by,  to  "Fisher's  Index  Num- 
ber," 242;  defense  of  Day's  index 
number  by,  316;  Formula  6023 
approved  by,  486;  cited,  313,  314, 
316,  326,  328,  331,  336,  340,  343, 
366,  384,  410,  411,  430,  486,  520. 

Pierson,  N.  G.,  objections  to  index 
numbers  quoted,  1;  time  reversal 
test  first  used  by,  82;  cited,  49,  117, 
224. 

Pigou,  A.  C.,  mention  of  Formula  353 
by,  241-242;  ideal  index  number 
approved  by,  482;  cited,  366,  519. 

Price  relative,  3,  16. 

Prices,  dispersion  of  individual,  11-14; 
errors  in  weights  less  important  than 
in,  447-449.. 

Probability  system  of  weighting, 
379-381. 

Probable  error  of  index  number,  341; 
Kelley's  method  of  measuring,  430— 
431;  of  Formula  353,  225-229;  deri- 
vation of,  of  13  formulae,  407-408. 

Product,  joint  error  expressible  by, 
88-90. 

Proportionality,  as  test  of  index  num- 
ber of  prices,  420-423. 

Purchasing  power,  an  index  number  of, 
377. 

Quantities,    dispersion    of     individual 

prices  and,  11-14. 
Quantity  relatives,  dispersion  of,  110- 

111. 
Quartets,    145;   arranging  of  formulae 

in,  145-147;  list  of,  164-170. 
Quotient,   joint   error  expressible   by, 

88-90. 

Ratio,  price,  3,  78;  quantity,  72-73, 

78;  value,  74,  78. 
Ratio  chart  method  of  plotting,  25-27. 


Ratios,  index  number  average  of, 
rather  than  ratio  of  averages,  451- 
457. 

Rawson-Rawson,  use  of  factor  antith- 
eses by,  134;  Formula  52  approved 
by,  471;  cited,  368. 

Reciprocals,  use  of,  in  calculating 
simple  harmonic  average,  30. 

Rectification  of  formulae,  136;  by 
crossing  time  antitheses,  140-142; 
by  crossing  factor  antitheses,  142- 
144;  of  simple  arithmetic  and  har- 
monic by  both  tests,  145-149;  by 
crossing  weights,  184-196;  order  of, 
398-399. 

Reversal  tests.   See  Tests. 

Samples,  use  of,  in  measuring  price 
movements,  330-331;  influence  of 
assortment  of,  331-334;  number  of, 
336-340. 

Sauerbeck,  A.,  simple  arithmetic 
approved  by,  459,  466;  cited,  111, 
117,  317,  342,  345,  349,  395. 

Schuckburgh-Evelyn,  G.,  simple  arith- 
metic approved  by,  458,  466. 

Scope  of  our  conclusions,  381-383. 

Scrope,  G.  Poulett,  cross  weight  aggre- 
gative formula  approved  by,  196; 
Formula  53  approved  by,  471;  For- 
mula 54  approved  by,  471;  Formula 
1153  approved  by,  483;  Formula 
9051  approved  by,  458. 

Sidgwick,  H.,  Formula  8053  approved 
by,  196,  487. 

"Skewness"  of  dispersion,  question 
concerning,  408-410. 

Speed  of  calculation  of  index  numbers, 
321-329. 

Splicing,  310-312;  as  applied  to  aggre- 
gative index  numbers,  427-428. 

Standard    deviation.     See    Deviation. 

Standard  Statistics  Corporation,  cited, 
438. 

Statist  (London),  simple  arithmetic  ap- 
proved by,  466;  cited,  29,  342,  345, 
349. 

Supplementary  series  of  formulae,  184. 

Tests,  reversal,  62-63;  commodity 
reversal,  63-64;  time  reversal,  64- 
65;  time  reversal,  illustrated  nu- 
merically and  graphically,  and  ex- 
pressed algebraically,  65-72;  factor 
reversal,  72 ;  simple  arithmetic  index 


526 


INDEX 


number  tested  by  factor  reversal, 
72-76;  factor  reversal,  illustrated 
graphically,  76-77;  error  revealed  by 
factor  reversal,  77-79;  factor  reversal 
analogous  to  other  reversal,  79-82; 
reversal,  as  finders  of  formulae,  118- 
135;  rectifying  formulae  by,  136-183; 
importance  of  conformity  to,  by  first- 
class  index  numbers,  268;  so-called 
circular,  270-295  (see  Circular  test) ; 
triangular,  295. 

Time  antithesis.   See  Antithesis. 

Time  reversal  test.    See  Tests. 

Time  studies  for  calculating  index 
numbers,  321-325. 

Times  Annalist,  cited,  460. 

United  States  Bureau  of  Labor  Statis- 
tics, aggregative  weighted  by  base 
year  values  approved  by,  471;  cited, 
53,  59,  240,  335,  341,  342,  344,  346, 
363,  369,  437,  438. 

United  States  Bureau  of  Standards, 
cited,  225. 

Value,  ratio.   See  Ratio,  value. 

Wages,  adjustment  of,  by  index  num- 
bers, 368,  460. 

Walras,  L.,  cited,  296. 

Walsh,  C.  M.,  cited,  29,  35,  40,  59,  60, 
121,  207,  255,  326,  328,  366,  408,  458, 
459,  519,  520;  on  use  of  simple  mode 
average,  39;  on  aggregative  form  of 
index  number,  42;  importance  of 
time  reversal  test  recognized  by,  82; 
idea  of  type  bias  expressed  by,  117; 
use  of  time  reversal  test  by,  134; 
cross  weight  aggregative  formula 
approved  by,  196;  reference  by,  to 
Formula  353,  241,  242;  on  the  so- 
called  circular  test,  295-296;  For- 


mula 22  approved  by,  468;  Formula 
1123  approved  by,  483;  ideal  index 
number  approved  by,  482;  Formula 
1153  and  1154  approved  by,  483; 
Formulae  2153  and  2154  approved 
by,  484. 

War  Industries  Board,  weighted  index 
number  of,  44,  342-343;  cited,  14, 
216,  262,  333,  334,  336,  339,  340, 
344,  410. 

Weighting,  6-8;  just  basis  for,  43-45; 
by  base-  year  values  or  by  given  year 
values,  45-53;  two  intermediate 
(hybrid)  systems  of,  53-56;  only 
two  systems  of,  for  aggregative  type 
of  index  number,  56-57;  history  of, 
59-60;  additional  systems  of,  61; 
bias  in,  91-94;  influence  of,  439  ff.; 
simple  and  cross,  compared,  443- 
444;  errors  in,  less  important  than 
in  prices,  447-449;  the  best  system 
of,  449-450. 

Westergaard,  idea  of  circular  test 
propounded  by,  295;  simple  geo- 
metric approved  by,  459,  468. 

Withdrawal  or  entry,  as  test  of  index 
number  of  prices,  420-423. 

Wood,  George  H.,  "Some  Statistics 
of  Working  Class  Progress  since 
1860"  by,  438. 

Young,  Allyn  A.,  estimate  of  Formula 
353  by,  242;  probability  system  of 
weighting  of,  379-380;  "  ideal "  index 
number  approved  by,  482;  cited, 
366,  520. 

Young,  Arthur,  Formula  9001  ap- 
proved by,  458,  487;  cited,  43,  45. 

Zizek,  Franz,  Statistical  Averages  by, 
438. 


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